Alexandra Destrez
email: alexandra.destrez@agrosupdijon.fr

Pauline Grimm

Véronique Julliand

Affiliations

Dietary-induced modulation of the hindgut microbiota is related to behavioral responses during stressful events in horses

Keywords: behavior, diet, microbiota, stress, horses

Introduction

The bidirectional communication between the central and the enteric nervous system named the gut-brain axis has been widely recognized [START_REF] Collins | The interplay between the intestinal microbiota and the brain[END_REF][START_REF] Cryan | Mind-altering microorganisms: the impact of the gut microbiota on brain and behaviour[END_REF][START_REF] Tengeler | Relationship between diet, the gut microbiota, and brain function[END_REF]. Over the past decade, fundamental studies emphasized the importance of the gut microbiota in influencing this communication, revealing a microbiota-gut-brain axis with two-way interactions between the microbiota and stress behavior [START_REF] Cryan | Mind-altering microorganisms: the impact of the gut microbiota on brain and behaviour[END_REF][START_REF] Foster | Stress & the gut-brain axis: regulation by the microbiome[END_REF]. The gut microbiota has been implicated in a variety of stress-related conditions including anxiety, depression and irritable bowel syndrome based on animal studies or correlative analysis in patient populations [START_REF] Foster | Stress & the gut-brain axis: regulation by the microbiome[END_REF]. For instance, human patients with depression exhibited modified and less diverse fecal microbiota signatures than controls [START_REF] Liu | Similar fecal microbiota signatures in patients with diarrhea-predominant irritable Bowel syndrome and patients with depression[END_REF].

In lab animals altered gut microbiota profiles have been associated with changes in stressrelated behavior and exacerbation of the hypothalamic-pituitary-adrenal axis reactivity [START_REF] Vuong | The microbiome and host behavior[END_REF]. As an example, stress hormones (plasma ACTH and corticosterone) elevation in response to restraint stress was higher in germfree mice than in conventional mice [START_REF] Sudo | Postnatal microbial colonization programs the hypothalamic-pituitary-adrenal system for stress response in mice[END_REF]. In open-field test (i.e. test of spontaneous activity), germfree rodents were more anxious and active than conventional rodents [mice: 8, rats : 9].

Bearing in mind this influence of the gut microbiota on the central nervous system, there is great temptation for changing behavior by modulating gut microbiota structure and function.

It is largely recognized that dietary modulation is a major and easy way for inducing changes in the composition of the gut microbiota [START_REF] Scott | The influence of diet on the gut microbiota[END_REF]. Furthermore there is scientific evidence supporting a causal relationship between overall diet quality and occurrence of depression and anxiety [START_REF] Jacka | Nutritional psychiatry: where to next?[END_REF]. For example, mice on high-sucrose diet displayed less anxiety than mice on high-fat diet in three stressful tests [START_REF] Jørgensen | A possible link between food and mood: dietary impact on gut microbiota and behavior in BALB/c mice[END_REF]. However, biological pathways and targets that mediate the associations between diet, gut microbiota and anxiety are not clearly identified. This is why we investigated in a earlier study to what extent behavioral changes were associated with changes of the microbiota through an alimentary change in horses [START_REF] Destrez | Changes of the hindgut microbiota due to high-starch diet can be associated with behavioral stress response in horses[END_REF]. Far from their natural living, most horses are submitted to stressful conditions in terms of housing, feeding or training that have been associated with the occurrence of altered behavioral responses or even of diseases such as gastric ulcers, colic or laminitis [START_REF] Bell | Equine gastric ulcer syndrome in adult horses: a review[END_REF].

Our earlier study indicated that the gut microflora and the horse's brain might have a bidirectional relationship because the animal's response in a novelty and a social test varied with diet [START_REF] Destrez | Changes of the hindgut microbiota due to high-starch diet can be associated with behavioral stress response in horses[END_REF]. More research is needed to evaluate the impact of the dietary stress factor on particular bacterial taxa and their interactions with behavioral changes. Hence, the present study aimed to investigate the link between changes in behavior during stressful events and changes in microbial composition and activity of the colonic ecosystem in horses. In view of our previous study, another behavioral test known to be more stressful for animals than a novelty test was used. In addition to bacteria enumeration, we determined bacterial richness, diversity and structure at family and genus level as well as pH.

Material and methods

The protocol was approved by the Committee on the Ethics of Animal Experiments of Grand Campus Dijon (registration number: 105; April 05, 2013).

Subjects and facilities

Six adult crossbred geldings fitted with cannulas in the right ventral (RV) colon were used.

The barrel of each fistula (in the cecum and in the right-ventral colon) was 15 cm long and 2.25 cm internal diameter. Surgery procedure used to set fistulas was the same as that described by [START_REF] Drogoul | Feeding ground and pelleted hay rather than chopped hay to ponies: 1. Consequences for in vivo digestibility and rate of passage of digesta[END_REF], and was performed at least five years ago. Hence, the effect of fistula on behavior was assumed to be negligible. Horses were aged 13-21 years and weighing 474 to 517 kg. They were housed in 3.3 × 4 m stalls which contained an automatic waterer (water provided free choice), a plastic feed bucket for hay, a plastic feed bucket for barley with a trace mineral and NaCl block, and wood shavings (Copeaux Classic, Thierwhol, Retteinmaier, France) as bedding over rubber stall mats. Horses were fed at 08:00 and 17:00; stalls were cleaned and bedding replaced each morning. The horses were released daily in a small dry paddock (10 × 10 m) for 2h. Horses were exercised six times a week (except days of digestive collection or behavioral tests) in an automatic walker 1h per day at 4-6 km/h. Their vaccinations against tetanus and influenza (ProteqFlu TE, Merial) and their de-worming (Equest Oral Gel (moxidectin), Fort Dodge Animal Health) were updated before the start of the experiment.

Experimental design and diets

The horses were submitted to a longitudinal experiment composed of 8 weeks and separated in 2 periods, each associated with a diet. During H diet (30 days), horses were fed a high-fiber diet composed of 100% of hay (H diet; 2.1 kg of Dry Matter (DM)/day/100 kg of Body Weight (BW)). Then, during a period of gradual transition (over five days), they were submitted to a modification of diet from the H diet to a progressive low-fiber and high-starch diet (from 90% hay and 10% barley to 60% hay and 40% barley in 4 days). During HB diet (23 days), horses were fed a diet composed of 56% of hay and 44% of barley (HB diet; 1.4 kg of DM/day /100 kg of BW with 0.8 kg of DM/day/100kg of BW hay). All the diets were formulated to be iso-energetics and to meet 105% of the energy requirements for horse subjected to a very light work [START_REF]Alimentation des chevaux[END_REF]. Both diets were offered in two equal meals per day. Every period, the body weight of each horse was assessed visually through a body condition score and horses were weighed.

Recent data (not published yet) in horses showed that 6-weeks after a change from a highstarch diet to a hay diet the hindgut microbial ecosystem was still statistically different from its status under the hay diet (Grimm & Julliand, personal data). Thus, we hypothesized that the modulation of the hindgut microbiota after a high-starch diet would last more than 4 weeks and we chose to observe all the 6-fistulated horses during hay diet (their usual diet) then during the high-starch diet. The timing of the experimental procedures is described in Fig. 1.

Intestinal samples collection and microbiota analyses

Colonic contents were collected 4h after the morning meal the 27 th day of the H diet and the 20 th of the HB diet (Fig. 1). The colonic contents were obtained by gravity via the cannulas.

An aliquot of the unfiltered contents (solid and liquid phase) was sampled for molecular biology analysis (1 mL, storage at -80 °C until analysis). Another aliquot of unfiltered content was also sampled in a container filled to capacity (to avoid the presence of oxygen) for microbiological analyses performed immediately after the sample collection.

Bacteria functional group analysis

Total anaerobic bacteria and cellulolytic, amylolytic and lactate-utilizing bacteria were enumerated in the colon using conventional anaerobic culture techniques. The content of colon (1mL) was serially diluted in a mineral solution [START_REF] Bryant | Cultural methods and some characteristics of some of the more numerous groups of bacteria in the bovine rumen[END_REF] and then inoculated in roll tubes on specific media under continuous flow of CO2 [START_REF] Hungate | Chapter IV A roll tube method for cultivation of strict anaerobes[END_REF]. Concentrations of total anaerobic bacteria (dilutions representing 10 -5 , 10 -6 , or 10 -7 mL of intestinal contents) were determined on a non-selective modified complete agar medium [START_REF] Leedle | Differential carbohydrate media and anaerobic replica plating techniques in delineating carbohydrate-utilizing subgroups in rumen bacterial populations[END_REF][START_REF] Julliand | Identification of ruminococcus flavefaciens as the predominant cellulolytic bacterial species of the equine cecum[END_REF] after 48 h of incubation at 38°C. Concentrations of lactate-utilising bacteria (dilutions representing 10 -3 , 10 -4 , or 10 -5 mL of intestinal contents) were determined on a selective medium [START_REF] Mackie | Enumeration and isolation of lactate-utilizing bacteria from the rumen of sheep[END_REF] after 48h of incubation at 38°C. Concentrations of amylolytic bacteria (dilutions representing 10 -4 ,10 -5 and 10 -6 , mL of intestinal contents) were determined on a modified selective medium containing 1% (w/v) soluble starch as the main energy source [START_REF] Lowe | Growth of anaerobic rumen fungi on defined and semi-defined media lacking rumen fluid[END_REF], after 72 h of incubation at 38°C. Most probable number [START_REF] Clarke | A simple and versatile micro-computer program for the determination of 'most probable number[END_REF] of cellulolytic bacteria (dilutions representing 10 -5 , 10 -6 , and 10 -7 mL of intestinal contents) were determined using a modified broth medium [START_REF] Julliand | Identification of ruminococcus flavefaciens as the predominant cellulolytic bacterial species of the equine cecum[END_REF][START_REF] Halliwell | The cellulolytic activity of pure strains of bacteria from the rumen of cattle[END_REF] after 15 d of incubation at 38°C. Finally, all the bacteria concentrations were turned into decimal logarithms (log 10 ).

Bacterial 16S rDNA extraction, sequencing and bioinformatical analysis

Total DNA was extracted from 0.25 g RV colonic content as described by Yu and Morrison [START_REF] Yu | Improved extraction of PCR-quality community DNA from digesta and fecal samples[END_REF]. The quantity of obtained DNA was assessed on a spectrophotometer (Eppendorf, Hamburg, Germany) and its purity was evaluated by calculating A260/280 ratios to verify contamination by proteins (ratio values between 1.6 and 2 were acceptable for nucleic acid extractions from digestive samples). DNA samples were then frozen at -80°C. The V3-V4 hypervariable region (459 bp in E. Coli) of the bacterial 16S ribosomal RNA gene (16S rRNA, 1542 bp in E. Coli) was amplified by PCR using the forward and reverse primers F343 (CTTTCCCTACACGACGCTCTTCCGATCTACGGRAGGCAGCAG) (adapted from [START_REF] Liu | Short pyrosequencing reads suffice for accurate microbial community analysis[END_REF]) and R784 (GGAGTTCAGACGTGTGCTCTTCCGATCTTACCAGGGTATCTAATCCT) (adapted from [START_REF] Andersson | Comparative analysis of human gut microbiota by barcoded pyrosequencing[END_REF]). The PCR mix was constituted of 10 ng of DNA, 5µL of buffer (MP Biomedicals, Illkirch-Graffenstaden, France), 1 µL of dNTP mix, 0.5µL of Taq polymerase (5 U/µL MTP TM Taq DNA Polymerase, Sigma, Saint-Louis, Missouri, USA), 1.25 µL (20 µM) of forward and reverse primer and was completed to 50 µL with sterile water. The conditions of PCR were: one cycle at 94 °C during 1 min, followed by 30 x (94 °C during 1 min, 65 °C during 1 min and 72 °C during 1 min). The amplification was followed by a melting program (72°C during 10 min). To verify the correct amplification of the V3-V4 region, the DNA samples were electrophoresed on a 2% agarose gel. The PCR products (4µL) were applied per well and electrophoresed for 1h15 at 45°C using a fixed voltage of 70 V on a Mupid One system. After purification with magnetic beads amplicons were submitted to a second PCR aiming at ligate Illumina adapters and an index allowing the identification of the sample. The PCR mix was the same than for the first PCR with a forward primer (AATGATACGGCGACCACCGAGATCTACACTCTTTCCCTACACGAC) and a reverse primer(CAAGCAGAAGACGGCATACGAGAT-Index-GTGACTGGAGTTCAGACGTGT).

The conditions of PCR were similar to the previous ones except 12 cycles of elongation were performed. PCR products were again purified using magnetic beads. The resulting PCR products were sequenced using an Illumina MiSeq run of 250 base paired-ends according to the manufacturer instructions (Illumina Inc., San Diego, CA). The quality of the run was checked using control libraries generated from the PhiX virus (Illumina PhiX control; Illumina Inc., San Diego, CA). Bioinformatic analyses were performed using the FROGS pipeline (Find Rapidly OTU with Galaxy Solution), which associates several tools to work on the file of sequences and obtain both the abundance table of operational taxonomic units (OTUs) and their taxonomic affiliation [START_REF] Escudie | find, rapidly, OTUs with galaxy solution[END_REF]. OTUs which are not present in at least 2 samples, or whose abundance was less than 5x10 -5 , were removed. Remaining OTU (operational taxonomic unit) were then aligned to the silva123 16S data base [START_REF] Pruesse | SILVA: a comprehensive online resource for quality checked and aligned ribosomal RNA sequence data compatible with ARB[END_REF] using BLAST and were affiliated to the highest taxonomic rank possible. The abundance table and the associated multi-hit list were created.

Relative abundance of the different families and genera was calculated. Richness (Number of species) and diversity (Shannon) indexes [START_REF] Peet | The measurement of species diversity[END_REF] were calculated from the abundance table using FROGS.

pH measurement

The pH of the colon was measured directly after the samples collection using the HI 1053 electrode (Hanna instrument, Lingolsheim, France) connected to an electronic pH-meter (CyberScan pH 510, Eutech Instrument, Strasbourg, France).

Behavioral analysis in the stressful tests

In order to examine the reactions specific to the stimulus studied, we attempted to test the horses' reaction in a context as neutral as possible: novelty and umbrella tests were thus performed in a barn adjacent to the stall where horses were living. Both tests were realized the same day, the 30 th days of the H diet and the 23 rd days of the HB diet (Fig 1 .). Novelty tests were realized on mornings and umbrella tests on afternoons.

Novelty test

The testing area (Fig. 2) consisted of a pre-test pen (waiting pen, 15 m 2 ) and a test arena (45 m 2 ) made of wooden walls covered with white sheets (2 m high). A sliding door allowed passage from waiting pen to the area [START_REF] Destrez | Long-term exposure to unpredictable and uncontrollable aversive events alters fearfulness in sheep[END_REF]. At the opposite side of the entrance, the test arena contained a bucket containing pellets (75 g of concentrate) and a closed black umbrella fixed on the wall (1.5 m high, the opening system of the umbrella was on the other side of the wall thanks to a hole in the wall). Three zones were determined in the test arena to measure the horse activity (Z1, Z2 and Z3). Between the 27 th day and the 30 th of the H diet, horses were submitted to a habituation procedure. On day one, the animals were individually allowed to freely explore the test arena for 5 min. On day two, each horse was led by a stockperson to the bucket containing the pellets and ate them. On day three, animals were allowed to eat pellets on their own. After the habituation procedure, horses were submitted to the novelty test on two occasions (Fig 1 .). Each horse entered the test arena where a novel object (H diet: a red fire extinguisher, 1 m high; HB diet: a gold cardboard display box, 1.5 m high) had been placed 20 cm in front of the food-filled bucket. The test lasted for 5 min. The behavior of the horse was recorded (using a SONY Handycam HDV Camera) and analyzed using the following behavioral categories: feeding (taking food into the mouth, chewing food), being vigilant (being immobile with head in an upright position, ears immobile and tail slightly raised), smelling the floor, interacting with the novel object (head and ears oriented towards the object whereas animal's nose is less than 10 cm from the object), blowing (high pitched 'whoosh' sound produced as the horse exhales through the nose) and moving (sum of the frequencies of being of the different zones Z1, Z2 and Z3). The order that animals entered in the testing area was randomly made.

Umbrella test

The day of the novelty test (in the same test arena), horses were submitted to the umbrella test on two occasions (Fig 1 .). Each horse entered the test arena and started to feed. After 30 s feeding, the sudden event occurred i.e. the black umbrella was open by an experimenter placed outside the pen. As soon as the umbrella was open, the test lasted for 5 min. The behavior of the horse was recorded (using a SONY Handycam HDV Camera) and analyzed using the following behavioral categories: feeding, being vigilant, smelling the floor, having a startle response (having a visible transient contraction of the shoulder and/or hindquarter of the animal occurring with a flexion of the legs or with the legs moving away from each other), blowing and moving. Times spent in the different zones of the test arena (Z1, Z2 and Z3) and latencies to feed after the sudden event were also recorded. The order that animals entered in testing area was randomly made.

Statistical analysis

Data were analyzed using XLSTAT software (version 2015.3.01.19097, Addinsoft, Paris, France). Since the animals were physiologically stable (adult, non-productive), restricted fed, and maintained in a controlled environment, time effect was assumed to be negligible [START_REF] Brossard | Ruminal fermentative parameters and blood acido-basic balance changes during the onset and recovery of induced latent acidosis in sheep[END_REF].

Diet effect on microbial parameters (concentration of bacteria functional group, relative abundance of bacterial families and genera, richness and diversity indexes) was analyzed using ANOVAs. Friedman test was used to analyze diet effect on horses' behaviors during novelty and umbrella tests. Spearman correlations (with Holm's method adjustment) were used to correlate microbial parameters and horses' behaviors during novelty and suddenness tests. The limit of significance was set at P < 0.05.

Results

Behavioral analysis

During novelty test, the frequency of blowing was significantly higher in HB diet than in H diet (P= 0.04, Table1). The other behavioral variables were not significantly different between H diet and HB diet (P>0.3, Table 1). During umbrella test, behavioral variables were not significantly different between H diet and HB diet (P>0.3, Table 1).

Intestinal microbiota analysis

Concentrations of amylolytic bacteria and of total anaerobic bacteria were significantly higher on HB diet than in H diet (respectively P= 0.005 and P= 0.05, Table 2). pH and the other concentrations of bacteria were not significantly different between the two diets (Table 2).

The number of species and Shannon index were not significantly different between diets (Table 2). The relative abundance of Succinivibrionaceae family was significantly higher in HB diet than in H diet (P= 0.04, Table 2). The relative abundance of the other bacteria families were not significantly different between the two diets (P> 0.1, Table 3).

Correlations between behavioral and physiological data

The adjusted P-values with Holm's method showed that behavioral and physiological data were not significantly correlated (adjusted P-values>0.5). In the following results, P-values of Spearman correlations without Holm's method adjustment are shown.

Novelty test

Concentration of amylolytic bacteria and frequency of blowing were positively correlated (P= 0.009, Table 3). The other behavioral variables and concentrations of bacteria were not significantly correlated (P>0.1, Table 3).

Behavioral variables and Species index were not significantly correlated (|r|<0.5; P>0.1).

Shannon index and duration of smelling the floor; Shannon index and frequency of moving were positively correlated (|r|>0.6; P<0.04). Shannon index and duration of feeding were negatively correlated (r=-0.8; P=0.01). The other behavioral variables and Shannon index were not significantly correlated (|r|<0.5; P>0.1).

Relative abundance of Succinivibrionaceae (genus: Succinivibrio) and frequency of blowing were positively correlated (P=0.02, Table 4). The other behavioral variables were not significantly correlated to colonic bacteria families relative abundance (P>0.1, Table 4).

Umbrella test

Behavioral variables and bacterial concentrations were not significantly correlated (P>0.1, Table 5). pH was negatively correlated with frequency of startle response (P=0.02, Table 5).

Behavioral variables and species number were not significantly correlated (|r|<0.6; P>0.3).

Behavioral variables and Shannon index were not significantly correlated (|r|<0.6; P>0.3).

Relative abundance of Ruminococcaceae (especially genus Ruminiclostridium and

Ruminococcaceae UCG-005) was positively correlated with duration of smelling the floor (P=0.05, Table 6). Relative abundance of Ruminococcaceae (especially genus Ruminiclostridium 5, Ruminococcaceae UCG-002 and Ruminococcaceae UCG-003) was negatively correlated with latency to feed (P=0.006, Table 6). Relative abundance of Prevotellaceae was positively correlated with latency to feed (P=0.04, Table 6). The other behavioral variables and bacteria family abundances were not significantly correlated (P>0.05, Table 6).

Discussion

In this study, we examined and found data supporting the association between the fluctuations of the colonic microbial ecosystem due to dietary changes and changes in behavior reflecting anxiety-like disorders in horses. One strength of our study included the use of live horses as model for their own species. Indeed interactions between the intestinal microbiota and the host are host specific [START_REF] Nicholson | Hostgut microbiota metabolic interactions[END_REF]. There is a marked inter-individual variability of the colonic bacterial composition, which could be a confounding factor. Using fistulated animals, this limitation could be resolved as we worked with the same individuals along the study. To increase the relevance of the findings, horses should be tested after different lengths of time eating each diet. Another study should be conducted to test the diets short-and long-term effects on the microbiota.

To study the variations of the colonic microbial ecosystem, we combined the analysis of bacterial composition (family) and microbial functionality of the colonic ecosystem. Recent data brought out the fact that studying the microbial metabolism could be as important and relevant as examining the gut microbiota composition in order to better understand the complexity of the dynamic relationship of microbiota, diet, and mental health [START_REF] Pizarro | Inter-relationship of the intestinal microbiome, diet, and mental health[END_REF]. Under dietary changes, the capability of bacteria to modify their metabolism could indeed be faster than their capacity to alter their composition. The two diets that we distributed to the horses modulated significantly the colonic microbiota in accordance with data reported in the literature. As expected [START_REF] Julliand | The impact of diet on the hindgut microbiome[END_REF], the different alterations demonstrated a lower fibrolytic capacity and a higher amylolytic capacity of the ecosystem when horses received higher starch intake. This change was reflected by the significant increase of amylolytic bacteria concentration in the colonic content of horses fed the high-starch diet compared the high-fiber diet.

Furthermore, the relative abundance of Succinivibrionaceae family increased with starch.

Bacterial species of Succinovibrio are included among the predominant amylose-, amylodextrin-and maltose-utilizing bacteria [START_REF] Kotarski | Starch hydrolysis by the ruminal microflora[END_REF][START_REF] Chesson | Polysaccharide degradation by rumen microorganisms[END_REF][START_REF] Stewart | The rumen microbial ecosystem[END_REF]. And it was even reported that Succinivibrio dextrinosolvens constitutes a high proportion of the cecal microbiome before the onset of laminitis [START_REF] Milinovich | Microbial events in the hindgut during carbohydrate-induced equine laminitis[END_REF]. Cellulolytic and amylolytic bacteria hydrolyse plant structural cellulose and starches respectively into simple sugars (cellobiose, glucose) which are further fermented by other intestinal microorganisms into pyruvate and then short chain fatty acids, lactate, and gases (CO2 and CH4). The types of fermentation metabolites produced from pyruvate in the hindgut of horses depend on the dietary environment [START_REF] Julliand | The impact of diet on the hindgut microbiome[END_REF] and can impact the fecal pH. Although we found no significant differences in the colonic content pH of horses fed high-starch or high-fiber diet, pH was negatively correlated with the frequency of startle response in umbrella test. Hence, when the colonic content pH decreased which can induce acidosis [START_REF] Julliand | Feeding and microbial disorders in horses: Part 3-Effects of three hay:grain ratios on microbial profile and activities[END_REF], the frequency of startle response increased which can reveal more anxiety in horses. Our data support the fact that a dietary factor can induce changes in behavior during stressful events. Likewise, during the novelty test our horses demonstrated a significantly higher frequency of blowing when they were fed the high starch compared to the hay diet. This frequency of blowing was positively correlated with the concentration of amylolytic bacteria and the relative abundance of Succinivibrionaceae family. Blowing in horses may be considered as an alert or alarm type of behavior and may be associated with anxiety [START_REF] Bagshaw | Behavioral and physiological effect of orally administered tryptophan on horses subjected to acute isolation stress[END_REF]. In our study, the horses seemed to be more over reactive in the novelty test when fed the high starch diet than the hay diet. However, they did not show significant behavioral differences during the umbrella test. During this test, all the horses stopped eating few seconds and some of them showed a startle response in reaction to the sudden event, with no difference between the diets. This lack of difference may be due to a ceiling effect, because all the horses exhibited very strong responses to the sudden opening of the umbrella. This hypothesis was also suggested in calves [START_REF] Boissy | Behavioural reactivity affected by chronic stress: An experimental approach in calves submitted to environmental instability[END_REF] and in lambs [START_REF] Destrez | Long-term exposure to unpredictable and uncontrollable aversive events alters fearfulness in sheep[END_REF]. In addition, according to the appraisal theories of cognitive psychology of Scherer to assess emotions [START_REF] Scherer | On the sequential nature of appraisal processes: indirect evidence from a recognition task[END_REF], the evaluation of suddenness is the most automatic evaluative process. It can be argued that suddenness caused specific reflex responses, making it difficult to discriminate the variation in sentience and decision making between individuals.

A recurring criticism in studies examining the link between colonic microbiota and anxietylike disorders is that data do not permit the elucidation of whether an altered microbiota are the cause or consequence of behavioral changes. As each individual in our study was submitted to the same controlled management parameters in terms of housing, bedding, exercise, etc. except for the diet, we could hypothetically assign the observed behavioral changes during stressful tests to the alterations of the microbial ecosystem due to dietary modifications. We also found positive correlations between the duration of smelling the floor or the frequency of moving during novelty test and the diversity of colonic bacteria; and between the duration of smelling the floor during umbrella test and the relative abundance of Ruminococcaceae. Oppositely, bacterial diversity was negatively correlated with the duration of feeding during novelty test. Smelling the floor and moving may reflect an exploratory behavior in horses in a calm situation [START_REF] Wolff | Experimental tests to assess emotionality in horses[END_REF]. Therefore individuals who were more reactive in our study spent less time smelling the floor but more time before eating and for eating, and concomitantly had a lower bacterial diversity within the colon. Ruminococcaceae is a family of fibrolytic and acid-intolerant bacteria that may decrease in abundance in response to dietary change (concentrate diet) and intestinal disease in horses [START_REF] Daly | Alterations in microbiota and fermentation products in equine large intestine in response to dietary variation and intestinal disease[END_REF]. The decrease in bacterial richness and diversity potentially reduces the resilience of the microbial ecosystem and can result in a higher susceptibility to dysbiosis. Epidemiologic data have suggested that colic could result from dysbiosis in the hindgut ecosystem as a result of dietary factors [START_REF] Julliand | The impact of diet on the hindgut microbiome[END_REF]. Other epidemiological data have cited the temperament, and especially the irritability and excitability of the horse as a risk factor for colic [START_REF] Gonçalves | Risk factors associated with colic in horses[END_REF].

Conclusion

Modulation of the gut microbiota by changing the diet seemed to change the horses' behavioral reactions in stressful situations. As in our previous study [START_REF] Destrez | Changes of the hindgut microbiota due to high-starch diet can be associated with behavioral stress response in horses[END_REF], behavioral responses of anxiety were related to hindgut microbiota indicators of a high-starch diet. The ability of diet to modulate the microbial structure and function of the gut shows the gut microbiota to be not only a sensitive organ but also one with great potential to modulate anxiety. More studies are needed to better understand the effect of diets on anxiety disorders in mammals.

Figure captions Fig 1 .

 captions1 Figure captions

Fig 2 .

 2 Fig 2. The testing area for novelty and umbrella tests. Three zones were defined: Z1, Z2 and

Fig. 1 .

 1 Fig. 1.

Fig. 2 .

 2 Fig. 2.

Table 1 .

 1 Observed behaviors (n= 6 horses) during H diet (100% hay) and HB diet (56% hay and 44% barley) in both standardized stressful tests (novelty and umbrella tests)

	Mean ± S.E.	H diet	HB diet	Friedman's test	P-value
	Novelty test
	Duration of feeding (s)	130.2 ± 35.8	122.2 ± 35.0	0.03	0.9
	Duration of being vigilant (s)	75.7 ± 26.8	85.0 ± 26.0	0.008	0.9
	Duration of smelling the floor (s)	29.3 ± 5.6	19.0 ± 8.0	0.8	0.4
	Duration of interacting with the	36.0 ± 13.4	49.4 ± 16.3	0.5	0.5
	novel object (s)
	Frequency of blowing	0.8 ± 0.8	3.4 ± 1.5	4.5	0.04
	Frequency of moving	8.8 ± 2.9	6.4 ± 1.2	0.08	0.8
	Latency to feed (s)	9.2 ± 1.4	12.4 ± 2.6	1.0	0.3
	Umbrella test
	Duration of feeding (s)	132.6 ± 38.9	97.4 ± 25.2	0.5	0.5
	Duration of being vigilant (s)	116.2 ± 39.5	142.6 ± 25.9	0.5	0.5
	Duration of smelling the floor (s)	23.6 ± 9.3	23.0 ± 9.2	0.01	0.9
	Frequency of startle response	0.6 ± 0.4	0.4 ± 0.2	0.06	0.8
	Frequency of blowing	3.6 ± 2.9	2.0 ± 1.3	0.01	0.9
	Frequency of moving	7.2 ± 2.5	7.6 ± 2.4	0.4	0.5
	Latency to feed (s)	12.8 ± 2.1	30.8 ± 22.6	1.1	0.3

Table 2 .

 2 Bacteria functional group, biochemical parameters and bacteria family (n= 6 horses) during H diet (100% hay) and HB diet (56% hay and 44% barley)

	Mean ± S.E.	H diet	HB diet	F	P-value
	Bacteria enumeration
	(CFU:Colony-forming unit)
	Concentration (log 10 CFU/g) of	4.7 ± 0.2	4.2 ± 0.2	3.5	0.09
	cellulolytic bacteria
	Concentration (log 10 CFU/g) of	5.6 ± 0.2	6.5 ± 0.2	13.7	0.005
	amylolytic bacteria
	Concentration (log 10 CFU/g) of	6.1 ± 0.3	6.9 ± 0.3	2.9	0.1
	lactate-utilizing bacteria
	Concentration (log 10 CFU/g) of total	6.9 ± 0.3	7.9 ± 0.3	5.3	0.05
	anaerobic bacteria
	pH	6.7 ± 0.1	6.9 ± 0.1	1.4	0.3
	Richness and diversity
	Species index	857 ± 50	845 ± 60	0.02	0.9
	Shannon index	5.5 ± 0.1	5.4 ± 0.1	0.5	0.5
	Family (relative abundance)
	Lachnospiraceae	21.7 ± 1.5	22.8 ± 1.9	0.2	0.7
	Ruminococcaceae	16.0 ± 1.0	20.0 ± 3.2	1.4	0.3
	Prevotellaceae	16.8 ± 2.5	10.4 ± 2.2	3.6	0.09
	Streptococcaceae	0.06 ± 0.02	4.0 ± 2.2	3.4	0.1
	Veillonellaceae	0.1 ± 0.05	1.5 ± 1.0	2.0	0.2
	Succinivibrionaceae	0.006 ± 0.006	1.3 ± 0.5	6.3	0.04
	Lactobacillaceae	0.3 ± 0.09	0.8 ± 0.3	3.2	0.1
	Fibrobacteraceae	0.2 ± 0.07	0.3 ± 0.2	0.1	0.8

Acknowledgments

The authors wish to thank staffs of the horse facility for technical assistance.


Paul-André Melliès
email: mellies@pps.jussieu.fr

Nicolas Tabareau
email: tabareau@pps.jussieu.fr

Free models of T -algebraic theories computed as Kan extensions

Keywords: Lawvere theories, PROs, PROPs, PROBs, operads, Kan extensions, distributors, enriched categories, free constructions, algebras, coalgebras. Contents

One fundamental aspect of Lawvere's categorical semantics is that every algebraic theory (eg. of monoid, of Lie algebra) induces a free construction (eg. of free monoid, of free Lie algebra) computed as a Kan extension. Unfortunately, the principle fails when one shifts to linear variants of algebraic theories, like Adams and Mac Lane's PROPs, and similar PROs and PROBs. Here, we introduce the notion of T -algebraic theory for a pseudomonad T -a mild generalization of equational doctrine -in order to describe these various kinds of "algebraic theories". Then, we formulate two conditions (the first one combinatorial, the second one algebraic) which ensure that the free model of a T -algebraic theory exists and is computed as an Kan extension. The proof is based on Bénabou's theory of distributors, and of an axiomatization of the colimit computation in Wood's proarrow equipments.

The tensor algebra

The investigation reported in this article is motivated by a basic and well-known problem in algebra, which we expose briefly here. Let k denote a commutative ring. To every k-module A is associated a particular k-algebra T A called its tensor algebra, defined as an infinite sum of tensorial powers:

T A = n∈N A ⊗n (1)
where we write A ⊗ B for the tensor product A ⊗ k B of two k-modules A and B, and A ⊗n for A ⊗ • • • ⊗ A taken n times. Note that we take the freedom here and onwards to consider this tensor product strictly associative.

In the language of category theory, one says that the construction T is functorial. This means that it defines a functor T : k-Mod -→ k-Alg from the category k-Mod of k-modules and k-module morphisms to the category k-Alg of kalgebras and k-algebra morphisms. Besides, this functor T is left adjoint to the forgetful functor

U : k-Alg -→ k-Mod (2)
which transports every k-algebra to its underlying k-module. The adjunction, noted

T ⊣ U
indicates that every morphism f : A -→ M from a k-module A to a k-algebra M factors uniquely as:

M A η G G f a a | | | | | | | | | T A h y y
where h is a k-algebra morphism and η is the canonical injection induced by Equation [START_REF] Barr | Toposes, triples and theories[END_REF]. This property states precisely that T A is the free k-algebra generated by the k-module A. Now, recall that a k-algebra M is defined as a k-module equipped with two morphisms,

k e -→ M m ← -M ⊗ M
called unit and multiplication, making the diagrams below commute:

M ⊗ M ⊗ M m⊗M G G M ⊗m M ⊗ M m M ⊗ M m G G M k ⊗ M e⊗M G G ∼ = 7 7 K K K K K K K K K K K K M ⊗ M m
So, a k-algebra is the same thing as a monoid object in the category k-Mod, seen as a monoidal category equipped with the familiar tensor product ⊗ of k-modules. Hence, the k-algebra T A is the free monoid object in the category k-Mod.

A basic problem in algebra

The existence of a free k-algebra T A for every k-module A should be contrasted to the fact, which is folklore among mathematicians, that there exists (in general) no free k-bialgebra for a given k-module. This deserves further explanation. First, recall that a k-cogebra k is a k-module equipped with two morphisms

k u ← -K d -→ K ⊗ K
called counit and comultiplication, making the diagrams dual to (3) (reverse the direction of arrows) commute. Hence, a k-cogebra is the same thing as a comonoid object in the monoidal category k-Mod. This defines a category k-Cog of k-cogebras and k-cogebra morphisms.

Then, a k-bialgebra H is a k-module equipped with a k-algebra and a k-cogebra structure, making the so-called Hopf compatibility diagrams commute:

H ⊗ H m G G d⊗d H d G G H ⊗ H H ⊗ H ⊗ H ⊗ H H⊗τ ⊗H G G H ⊗ H ⊗ H ⊗ H m⊗m y y k e G G d H d k ⊗ k e⊗e G G H ⊗ H H u 1
k e c c k H ⊗ H u⊗u G G m k ⊗ k m H u G G k 3
Here, τ denotes the symmetry H ⊗ H -→ H ⊗ H of the symmetric monoidal category k-Mod. Again, we provide a description of this compatibility through string diagrams

= = = =
Note that the categories k-Alg and k-Cog inherit the monoidal structure from the category k-Mod, and that a k-bialgebra may be seen alternatively as a monoid object in the category k-Cog of comonoid objects, and as a comonoid object in the category k-Alg of monoid objects. Now, let k-Big denote the category of k-bialgebras, and k-algebra and k-cogebra morphisms between them. It is folklore that the forgetful functor

U bij : k-Big -→ k-Mod (4)
does not have a left adjoint, for k an arbitrary commutative ring. This result states precisely that there exists no such thing as a free k-bialgebra.

In this article, we want to understand more conceptually what distinguishes the forgetful functor (2) which has a left adjoint, from the forgetful functor (4) which does not have a left adjoint. Moreover, we want to extract, in the good situations, a formula like (1) to compute the left adjoint, and hence the free construction, associated to the forgetful functor.

We carry out the investigation in Lawvere's functorial semantics, and take advantage of the extraordinary toolbox provided by categorical algebra, more specifically the concepts of algebraic theory and Kan extensions -which we explain now.

Lawvere theories

In his dissertation [START_REF] Lawvere | Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories[END_REF], Lawvere reformulates the familiar notion of algebraic theory (eg. of monoids, Lie algebras, etc.) as a category L with finite products, whose objects are the natural numbers: 0, 1, 2, . . . and in which the categorical product of k objects m 1 , . . . , m k is provided by their arithmetic sum m 1 + . . . + m k . As will become clear in the course of the article, the underlying idea is to represent every n-ary operation of the algebraic theory as a morphism n -→ 1 of the category L; and to encode the equational theory on these n-ary operations inside the composition law of the category L.

Then, an L-model in a category C with finite products (noted ×) is defined as a finite-product preserving functor

A : L - → C.
By "finite-product preserving functor", one means that the canonical morphism

A[m 1 + . . . + m k ] - → A[m 1 ] × . . . × A[m k ] (5)
is an isomorphism. This implies that the functor A(-) transports (up to isomorphism) the natural number n to the n-th power of the object A = A [START_REF] Barr | Toposes, triples and theories[END_REF]:

A[n] ∼ = A ×n .
Hence, the functor A transports (up to isomorphism again) every n-ary operation n -→ 1 of the algebraic theory to an n-ary operation on the object A:

A ×n - → A. (6)
For that reason, A is called the underlying object of the L-model, and the morphism ( 6) is called the interpretation of the n-ary operation n -→ 1. The terminology is justified by the fact that the L-model A is characterized (up to natural isomorphism) by its underlying object A and the interpretation of every n-ary operation of the algebraic theory L.

Every algebraic theory L and category C with finite products define together a category Model(L, C) whose objects are the L-models in the category C, and whose morphisms A -→ B are the natural transformations from the functor A to the functor B.

Kan extensions

An algebraic morphism between algebraic theories

j : L 1 -→ L 2
is defined as a finite-product preserving functor, such that j(1) = 1. Such an algebraic morphism induces a functor

U j : Model(L 2 , C) -→ Model(L 1 , C)
called the associated forgetful functor, which transports every L 2 -model

L 2 B -→ C to the L 1 -model L 1 j -→ L 2 B -→ C
obtained by precomposing with the functor j. By way of illustration, consider the free category F op with finite products, generated by the category with one object: this defines an algebraic theory whose n-ary operations are precisely the n projection maps π 1 , . . . , π n : n -→ 1. The category F op may be defined more explicitly as the opposite to the category of finite sets [m] = {1, . . . , m} and functions between them. This particular algebraic theory F op is called trivial because a F op -model is characterized (up to natural isomorphism) by its underlying object: hence, the category Model(F op , C) of F op -models is equivalent to the category C itself.

Next, consider the algebraic theory M of monoids whose n-ary operations are the finite words (of arbitrary length) built on an alphabet [n] = {1, . . . , n} of n letters. By construction, the category Model(M, C) is equivalent to the category of monoids and monoid morphisms in the category C. Now, there exists a (unique) algebraic morphism j : F op -→ M, which transports the i-th projection π i ∈ F op (n, 1) to the one-letter word i ∈ M(n, 1). It is not difficult to check that the associated forgetful functor U j transports every monoid in the category C to its underlying object. This leads to Lawvere's elegant formulation of "free constructions" in functorial semantics, using Kan extensions. Recall that the left Kan extension of a functor A : L 1 -→ C along a functor f : L 1 -→ L 2 is defined as a functor, noted Lan j A, and a natural transformation:

C L 1 j G G A b b ~⇒ L 2
Lan j A @ @ @ @ inducing a bijection:

[Lan j A, B] ∼ = -→ [A, B • j] (7)
between sets of natural transformations, for every functor B : L 2 -→ C. Now, suppose that C is the category Ens of sets and functions, or to that purpose, any cartesian closed category with small colimits. Then, the left Kan extension along j of any functor L 1 -→ C exists, and a miracle happens: when j is an algebraic morphism, the left Kan extension transports a finite-product preserving functor A to a finite-product preserving functor Lan j A. In this way, every L 1 -model A is transported to a L 2 -model Lan j A, and the bijection [START_REF] Lack | Note on the construction of free monoids[END_REF] specializes to the following bijection:

Model(L 2 , C)(Lan j A, B) ∼ = -→ Model(L 1 , C)(A, U j B)
between sets of morphisms, for every L 2 -model B. The bijection is natural in B, which shows that the left Kan extension Lan j A is the free L 2 -model generated by the L 1 -model A, along the algebraic morphism j.

The free construction is functorial: the left Kan extension Lan j A defines a left adjoint functor to the forgetful functor:

Lan j ⊣ U j : Model(L 1 , C) → Model(L 2 , C).
This is precisely what happens when j : F op -→ M is the algebraic morphism from the trivial algebraic theory F op to the algebraic theory M of monoids. In that case, the functor Lan j transports every set A (identified here to a F op -model) to the free monoid Lan j A whose underlying set A * = (Lan j A) [START_REF] Barr | Toposes, triples and theories[END_REF] is the set of finite words on the alphabet A:

A * = n∈N A ×n . (8)
This well-known equation may be obtained by direct means, or deduced from a general "coend" formula for computing left Kan extensions in the category Ens -a formula which appears in [START_REF] Lane | Categories for the working mathematician[END_REF] and will be deduced in the course of the article from Bénabou's theory of "distributors".

Linear theories (PROs)

Let us compare the two formulas (1) and [START_REF] Lawvere | Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories[END_REF]. The analogy is striking: the two formulas appear to compute in the same way the free monoid object T A or A * generated by an object A in their respective categories k-Mod and Ens. There is a fundamental difference however: the finite product A ×n appearing in the definition of the free monoid A * is replaced by a tensor product A ⊗n in the definition of the tensor algebra T A. This motivates to adapt Lawvere's ideas, and to replace every "finite product" by a "tensor product" in all the definitions related to algebraic theories. So, a linear theory L (also called a PRO) is defined as a monoidal category whose objects are natural numbers: 0, 1, 2, . . . and in which the tensor product of k objects m 1 , . . . , m k is provided by their arithmetic sum m 1 + . . . + m k . An L-model in a monoidal category C is then defined as a monoidal functor L -→ C.

By "monoidal functor" one means a functor A equipped with a family of isomorphisms:

A[m 1 + . . . + m k ] - → A[m 1 ] ⊗ . . . ⊗ A[m k ]
satisfying a series of expected coherence properties.

As for algebraic theories, every linear theory L and monoidal category C induce together a category Model(L, C) whose objects are the L-models in C, and whose morphisms θ : A -→ B are the monoidal natural transformations from A to B, seen as monoidal functors. Recall that a natural transformation θ is monoidal when the expected coherence diagram

A[m 1 + . . . + m k ] G G θ m 1 +•••+m k A[m 1 ] ⊗ . . . ⊗ A[m k ] θm 1 ⊗•••⊗θm k B[m 1 + . . . + m k ] G G B[m 1 ] ⊗ . . . ⊗ B[m k ]
commutes, for every natural numbers m 1 , . . . , m k . It is important to notice at this point that the category Model(L, C) just defined coincides with the category Model(L, C) defined earlier for algebraic theories -when the tensor product of the two categories L and C happens to be a finite product. The reason is that (a) every finite-product preserving functor defines a monoidal functor equipped with the canonical family of morphisms ( 5) and (b) every natural transformation θ between finite-product preserving functors satisfies the coherence properties required of a monoidal natural transformation.

In order to carry on our running example based on monoids, we introduce here two linear theories N and ∆ playing the role of the two algebraic theories F op and M defined earlier. First, consider the free monoidal category N generated by one object: this defines the trivial linear theory in which the only morphisms are the identities on each object 0, 1, 2, . . . Then, define the linear theory of monoids as the category ∆ of augmented simplices in which a morphism m -→ n is a monotone function from the ordered set [m] = {1, . . . , m} to the ordered set [n] = {1, . . . , n}.

Just as for the algebraic theories, the category Model(N, C) is equivalent to the category C, and the category Model(N, C) is equivalent to the category of monoids and monoid morphisms in C. The careful reader will notice for instance that the expected notion of "homomorphism" between monoids is indeed captured by the notion of monoidal natural transformation -and not just of natural transformation -between ∆-models.

Then, a linear morphism j : L 1 -→ L 2 between linear theories is defined as a monoidal functor such that j(1) = 1. Just as in the case of algebraic theories, every linear morphism j : L 1 -→ L 2 induces a forgetful functor

U j : Model(L 2 , C) - → Model(L 1 , C)
which transports every L 2 -model B to the L 1 -model B • j obtained by precomposing with the functor j.

Monoidal Kan extensions

At this point, we are ready to tell quite the same story as for algebraic theories -at least when it comes to computing free monoids in the category C = k-Mod of k-modules. By definition of the category N as a free monoidal category, there exists a unique linear morphism

j : N - → ∆. (9)
Then, every k-module A : N -→ k-Mod (seen as a model of the trivial theory N) is transported by left Kan extension along j to a functor defined by the general formula [START_REF] Lane | Categories for the working mathematician[END_REF] mentioned earlier:

Lan j A : p → n∈N ∆(n, p) ⊗ A ⊗n
where the k-module ∆(n, p) ⊗ A ⊗n means the direct sum of as many copies of the k-module A ⊗n as there are elements in the hom-set ∆(n, p). It appears after close inspection that this functor Lan j A is monoidal, and thus defines a ∆-model, which coincides with the tensor algebra and free monoid T A generated by the object A in the category k-Mod. Note that monoidality of the functor Lan j A amounts essentially to an isomorphism

Lan j A(p + q) ∼ = Lan j A(p) ⊗ Lan j A(q)
for every pair of natural numbers p and q. The isomorphism itself derives from the bijection

∆(n, p + q) ∼ = n k=0 ∆(k, p) × ∆(n -k, q)
between the coefficients of each part, which expresses that every monotone function f : [n] -→ [p + q] decomposes as a pair of monotone functions

f 1 : [k] - → [p] and f 2 : [n -k] - → [q] for some natural number 0 ≤ k ≤ n.
This success with monoids is somewhat miraculous, and should not deceive us. We are facing a serious difficulty indeed: in contrast to what happens with algebraic theories, the very fact that a L 1 -model A defines a left Kan extension along a linear morphism j : L 1 -→ L 2 does not qualify the resulting functor Lan j A as the free L 2 -model generated by A. . . There are two reasons for that defect: first, nothing ensures that the functor Lan j A is monoidal, and thus, that it defines a L 2 -model; then, even when the functor Lan j A happens to be monoidal, nothing ensures that this functor Lan j A defines indeed what we expect: the free L 2 -model generated by the L 1 -model A along the linear morphism j. The reason is that the Kan extension property provides a one-to-one correspondence between the natural transformations Lan j A -→ B and the natural transformations A -→ U j B, for every L 2 -model B -whereas a correspondence between monoidal natural transformations is required.

At this point, it is critical to proceed conceptually, and to remember that the notion of Kan extension may be defined in any 2-category. Typically, the left Kan extension considered until now is computed in the 2-category Cat with

• categories as 0-cells,

• functors as 1-cells,

• natural transformations as 2-cells.

What our discussion establishes is that we need instead a monoidal left Kan extension, that is: a left Kan extension computed in the 2-category MonCat with

• monoidal categories as 0-cells,

• monoidal functors as 1-cells,

• monoidal natural transformations as 2-cells.

In this way, all the difficulties are reduced to the single question:

When is the left Kan extension of a monoidal functor A along a monoidal functor j, a monoidal left Kan extension?

By this, we mean that the left Kan extension exists in MonCat and is transported by the forgetful 2-functor MonCat -→ Cat to the left Kan extension computed in Cat. Everything works nicely when this lifting property occurs: indeed, the left Kan extension Lan j A defines in that case the free L 2 -model generated by the L 1 -model A along the linear morphism j.

Interestingly, this lifting property always occurs with algebraic theories: this is precisely what makes these particular theories so convenient to compute free constructions. More specifically, let CartCat denote the 2-category with

• categories with finite products as 0-cells,

• finite-product-preserving functors as 1-cells,

• natural transformations as 2-cells.

It appears that the 2-category CartCat is a full sub-2-category of the 2-category MonCat, in the expected sense. Moreover, the left Kan extension of a finite-product-preserving functor L 1 -→ Ens along a finite-product-preserving functor j : L 1 -→ L 2 always exists in CartCat (and thus in MonCat) and coincides with the left Kan extension computed in Cat.

We will see that the lifting property holds when one computes the left Kan extension T A = Lan j A of a k-module A along the linear morphism j : N -→ ∆. This explains why the formula (1) works, and computes indeed the free monoid T A in the category of k-modules.

But the lifting property does not occur in any situation: consider for instance the linear theory of comonoids, defined as the category ∆ op obtained by reversing the maps of the category ∆. It is well-known that in a category C with finite products (for instance Ens) every object defines a comonoid in a canonical and unique way; and that every morphism A -→ B is then a comonoid morphism. From this follows trivially that every object A in the category C, when seen as a comonoid, defines the free comonoid generated by the object A. This reformulated in categorical semantics says that the commutative diagram

C N g G G A X X u u u u u u ∆ op A e e K K K K K K
defines a monoidal left Kan extension along the linear morphism g. When C = Ens, this says that the monoidal left Kan extension of a set A : N -→ Ens (seen as a model of the trivial theory N) is the set A itself. On the other hand, the usual left Kan extension of A : N -→ Ens along g : N -→ ∆ op is defined by the classical formula

Lan g A : p → n∈N ∆(p, n) × A ×n .
Hence, the left Kan extension of A along g computed in Cat does not coincide with its monoidal left Kan extension computed in MonCat.

Symmetric theories (PROPs)

This brings us back to the discussion on free k-bialgebras which opened the article. Just like the notion of algebraic theory was replaced by the notion of linear theory, by shifting from categories with finite products to monoidal categories, the notion of linear theory may be replaced by the notion of symmetric theory (also called PROPs), by shifting from monoidal categories to symmetric monoidal categories.

The trivial symmetric theory S is then defined as the free symmetric monoidal category on one object, whose only morphisms are the bijections At this point, we are ready to recast Loday's observation (mentioned earlier) in the language of categorical semantics. Let SMonCat be the 2-category with the symmetric monoidal categories as 0-cells, the symmetric monoidal functors as 1-cells, and the monoidal natural functors as 2-cells. Every k-module A may be seen as a symmetric monoidal functor S -→ k-Mod, induces a left Kan extension Lan j (A) along the (unique) symmetric monoidal functor j : S -→ B. Unfortunately, and this is the whole point, this left Kan extension in Cat is not a left Kan extension in the 2-category SMonCat; otherwise, it would define a free construction for k-bialgebras, and a left adjoint to the forgetful functor U j mentioned in (4): and this would contradict Loday's observation.

T -algebraic theories

In this article, we follow the ideas developed by Lawvere in his work on equational doctrines [START_REF] Lawvere | Ordinal sums and equational doctrines[END_REF] and formulate the notion T -algebraic theory for a pseudomonad T on the 2-category Cat. The notion describes in a generic way the various kinds of algebraic, linear, symmetric or even braided theories -each kind of theory described by a particular pseudomonad T . The only property required of the pseudomonad T is that it distributes over the presheaf monad, in the sense of [START_REF] Cheng | Pseudo-distributive laws[END_REF][START_REF] Gambino | On the coherence law for pseudo-distributive laws[END_REF], so that it lifts to a pseudomonad on the bicategory of distributors.

• algebraic theories = free category with finite products,

• linear theories = free monoidal category,

• symmetric theories = free symmetric monoidal category,

• braided theories = free braided monoidal category,

• projective sketches = free category with finite limits.

Every such pseudomonad T induces a 2-category Cat T with T -algebraic categories as 0-cells, T -algebraic morphisms as 1-cells, and T -algebraic natural transformations as 2-cells -where we apply the following dictionary: Definition 1 (T -algebraic theory) Let T be a pseudomonad on Cat. We consider the 2category Cat T whose 0-cells are T -algebraic categories, 1-cells are T -algebraic functors and 2-cells are T -algebraic natural transformations -where the following dictionary has been used:

• T -algebraic category = pseudoalgebra of the pseudomonad T ,

• T -algebraic functor = pseudomorphism of pseudoalgebras,

• T -algebraic natural transformation = invertible algebraic 2-cell.

A T -algebraic theory is defines as a small T -algebraic category -that is to say whose class of objects is a set.

Remark that our definition of T -algebraic theory is quite generous, and does not try to characterize which algebraic theories should be accepted or rejected according to the rank of the pseudomonad T . This has to be opposed to the approach of Kelly and Power on the elaborated work on enriched Lawvere theories [START_REF] Kelly | Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads[END_REF][START_REF] Power | Enriched lawvere theories[END_REF].

Definition 2 (model of a T -algebraic theory) A model M of a T -algebraic theory L (L- model) in a T -algebraic category C is a T -algebraic functor A : L → C.
We construct a category Model(L, C) of L-models and T -algebraic natural transformations between them.

We can now reformulate the question raised by the computation of free models by the following:

When is a left Kan extension of a T -algebraic functor A along a T -algebraic functor j, a T -algebraic Kan extension ?

Algebraic distributors at work

Besides our interest in algebraic theories, this article is motivated by our fascination for the theory of distributors -and the desire to promote this elegant theory to larger circles of algebraists. The theory of distributors was introduced by Jean Bénabou in the late 1960s, and recast in the language of enriched categories by the Sydney School, around Max Kelly and Ross Street.

There are at least two different ways to see a distributor:

• as a generalized functor,

• as a module with several objects.

Depending on the point of view, a distributor is called a profunctor, or a module. Here, we stick to the terminology introduced by Jean Bénabou, who had in mind the analogy between distributors and distributions in functional analysis [START_REF] Bénabou | Les distributeurs[END_REF].

We have already mentioned the existence of a classical formula which enables to compute the left Kan extension of a functor f along a functor j in many situations of interest. The formula appears for instance in Chapter 10 on Kan extensions of the famous monograph devoted to categories by Saunders Mac Lane [START_REF] Lane | Categories for the working mathematician[END_REF].

It is folklore among category theorists that the theory of distributors explains this formula in a beautiful and pleasingly conceptual way. The main idea of the reconstruction is that every functor

f : A - → B
induces a pair of adjoint distributors f * ⊣ f * with source and target as below:

f * : A→ B f * : B→ A.
The left Kan extension of the functor f : A -→ C along the functor j is then computed by "composing" the two distributors j * : B→ A and f * : A→ C , then by "taking the representative" of the resulting distributor -that is a functor Lan j (f ) : B → C that satisfies

Dist(f * • j * , g * ) ∼ = Cat(Lan j (f ), g)
for every functor g : B → C. The terminology representative comes from the classical situation of an object representing a functor as described by Barr and Wells [START_REF] Barr | Toposes, triples and theories[END_REF].

Once understood precisely how left Kan extensions are computed using distributors, there remains to understand when the resulting Kan extensions can be T -algebraic. The aim of this article is to clarify when this situation holds. We herald briefly the two recipes of our construction :

the adjunction j * ⊣ j * is T -algebraic the T -algebraic distributor f * • j * : A→ C is "represented" by a T -algebraic functor
As we want our reasoning to be kept at an algebraic level and as we would like to explain the enriched case as well, we need to construct our argument by abstracting the properties of the bicategory Dist of distributors. For that reason, we introduce the formalism of proarrow equipment that captures the essence of Bénabou's construction.

Proarrow equipments

Basic definitions

The notion of proarrow equipment is an axiomatization, introduced by Richard Wood [START_REF] Wood | Abstract proarrows i[END_REF][START_REF] Wood | Proarrows ii[END_REF], of the homomorphism of bicategories between Cat and Dist. We used here the original notations of Wood K and M for the two bicategories of concern.

Definition 3 (proarrow equipment)

A proarrow equipment is a morphism of bicategories (-) * : K → M that satisfies the three axioms:

1. the object of M are those of K and (-) * is an identity on objects morphism, 2. (-) * is locally fully faithful, i.e. K(f, g) ∼ = M(f * , g * );

3. for every 1-cell f of K, f * has a right adjoint f * .

Let us mention some examples of proarrow equipments: In the proarrow equipment Cat → Dist, when a functor f : C → D satisfies "f * is a retract of f * ", then we have

• K = V-Cat et M = V-Dist,
B(f c, f c ′ ) ∼ = A(c, c ′ ),
that is to say f is fully faithful. The property becomes here a definition.

Definition 4 (fully faithful)

We say that a morphism f :

A → B of K is fully faithful iff f * is a retract of f * .
The notion of representability in a proarrow equipment comes from the corresponding notion for a functor. Usually, given a functor R : D → C, we say that an object Lc represents the functor C(c, R(-)) as soon as

C(c, R(-)) ∼ = D(Lc, -).
It is well-known that when the functor C(c, R(-)) is representable for every object c of C, the functor R has a left adjoint [START_REF] Barr | Toposes, triples and theories[END_REF]. We extend this notion to the setting of proarrow equipment.

Definition 5 (representative)

A morphism g : B → C of K is a representative of a morphism f : B → C of M when there is a one-to-one correspondence M(f, h * ) ∼ = K(g, h)
for every h : B → C in K.

The notion of representative allows to find the "closest" morphism in K of a morphism in M. What we want to do is thus to compute a Kan extension in K by going throw M via the equipment, and then by going back to K via the representative.

Yoneda situations

We now give a recipe to compute the representative. It is well known that the computation of the representative of a presheaf on a category C is induced by the existence of some colimits in this category C.

To provide an abstract point of view on this colimit computation, we introduce the concept of Yoneda situations in a proarrow equipment. This situation allows to describe abstractly the morphisms of Cat from a category C to a category C that can be seen as restrictions of the Yoneda embedding from C to presheaves overt C. Definition 6 (Yoneda situation) A morphism y : C → C of K is said to be a Yoneda situation when

• y is fully faithful,

• y * is pseudomonic with respect to K, that is to say the functor

y * • (-) * : K 1 (A, C) → M 1 (A, C) is fully faithful for every object A of K.
This definition of a Yoneda situation allows to give a general notion of C-cocompleteness on a proarrow equipment. A famous case of C-cocompleteness for the proarrow equipment Cat → Dist is obtained when C is the category of presheaves on C and y is the Yoneda embedding. We then say that the category C is said to be total. In that case, the category is not only cocomplete but also complete.

Let us take a morphism f : B → C in the bicategory M. Suppose that C is C-cocomplete for the Yoneda situation y : C → C and that f factories through y * in the sense that

B f -→ C = B f * -→ C y * -→ C. Proposition 1 The morphism colim •f is a representative of f .
Proof : The proof follows from the following cascade of equivalences

M(f, g * ) = M(y * • f * , g * ) f = y * • f * ∼ = M(y * • f * , y * • y * • g * ) y is fully faithful ∼ = K(f , y • g) y * is pseudomonic with respect to K ∼ = K(colim •f , g) colim is the left adjoint of y
Let us explain this construction in the paradigmatic equipment Cat → Dist. We use here factorization system on Cat given by final functors and discrete fibrations [START_REF] Street | The Comprehensive Factorization Of A Functor[END_REF]. Thus, every diagram f : J → C can be seen as a presheaf ϕ given by the decomposition

J f -→ C = J f 1 -→ Elt(ϕ) f 2 -→ C
where f 1 is a final functor and f 2 is a discrete fibration. We now follow the notion introduced by Kelly of F-cocompleteness, and consider a cocomplete category C cocomplete for a given class F of categories, called the indices, containing the category 1. This means that there is an adjunction

C colim 3 3 ⊥ C y
between the category C of diagrams whose base is in F and the category C. The functor y sends every object c of C to the constant diagram y : C → (1 → C). Using the Yoneda lemma, we know that y is fully faithful and that the functor y * • (-) * is equal to the identity. We directly deduce that y * is pseudomonic with respect to Cat. Thus, the functor y is a valid Yoneda situation.

We can now show that the notion of representative enables to compute left Kan extension in the bicategory K.

Proposition 2 Let f : A → C and j : A → B be two morphisms of K. The representative of the morphism f * • j * (if it exists) is the left Kan extension of f along j in K.

Proof : As f * ⊢ j * , the morphism f * • j * is the left Kan extension of f * along j * in M. Note Lan j f its representative. For every g : B → C in K, we have the following cascade of bijections

K(Lan j f, g) ∼ = M(f * • j * , g * ) Lan j f represents f * • j * ∼ = M(f * , g * • j * ) f * • j * left Kan extension in M ∼ = M(f * , (g • j) * ) functoriality of (-) * ∼ = K(f, g • j)
(-) * is fully faithful which establishes that Lan j f is the left Kan extension of f along j in K.

Pseudomonads in proarrow equipments

To describe the notion of law algebra in our proarrow equipment, we need to extend the notion of pseudomonads in that setting. We follow the tradition on distributive laws, and particularly the work of Nicola Gambino [START_REF] Gambino | On the coherence law for pseudo-distributive laws[END_REF] where he defines morphisms of pseudomonads. Definition 8 (pseudomonad on a proarrow equipment) A pseudomonad on a proarrow equipment (-) * : K → M is given by • a pseudomonad T K on the bicategory K;

• a pseudomonad T M on the bicategory M;

• a pseudo natural transformation h :

T M • (-) * → (-) * • T K noted T M (-) * (-) * T K
making ((-) * , h) a morphism of pseudomonads from T K to T M in the sense that the following equalities hold

T M T M (-) * (-) * T K = T M T M (-) * (-) * T K (-) * (-) * T K = (-) * (-) * T K
A typical example of pseudomonads on a proarrow equipment is given by the pseudomonad that distribute with Yoneda. Those pseudomonads induce naturally a pseudomonad on the proarrow equipment Cat → Dist. This situation has been studied in details by Francisco Marmolejo [START_REF] Marmolejo | Distributive laws for pseudomonads[END_REF], and then by Eugenia Cheng, Martin Hyland and John Power [START_REF] Cheng | Pseudo-distributive laws[END_REF]. A similar work can be done in the enriched setting. All pseudomonads that will be consider bellow, in the setting of T -algebraic theories, share the property of distributing with Yoneda.

The notion of pseudomonad on a proarrow equipment allows to state that the two pseudomonads T K and T M are compatible with (-) * in the following sense.

Proposition 3

The morphism (-) * : K → M lifts to a morphism (-) T * : Lax-T K -Alg → Lax-T M -Alg (that we will abusively still note (-) * ) that restricts, through the inclusion functor from pseudoalgebras to lax algebras, to a morphism (-) T * : Ps-T K -Alg → Ps-T M -Alg. Those morphisms make the following diagram commute

Ps-T K -Alg (-) T * G G Ps-T M -Alg Lax-T K -Alg U K (-) T * G G Lax-T M -Alg U M K (-) * G G M
where U K and U M are the usual forgetful morphisms.

Proof : The diagrams are easy to check and this has been done in the work of Nicola Gambino [START_REF] Gambino | On the coherence law for pseudo-distributive laws[END_REF].

General setting

As sketched in the introduction, we want to compute free models by Kan extension on algebraic theories, PROs, PROPS, or even projective sketches. This has lead us to the definition of a common setting for all those kinds of theories that we call T -algebraic theories for a pseudomonad T on the proarrow equipment Cat → Dist. In the following, we use the same terminology for the straightforward generalization of T -algebraic theories to a pseudomonad on any proarrow equipment.

Our main result

We will know state the main result of this paper at the level of proarrow equipments, before we explain in more details all the needed hypotheses. From now on, we work with a pseudomonad (T K , T M , h) on a proarrow equipment (-) * : K → M. As we have seen with Proposition 2, the left Kan extension of a T K -algebraic morphism f : A → C along a T K -algebraic morphism j : A → B in K is obtained by precomposing f * with the right adjoint j * and then by taking the representative of resulting morphism. It remains to clarify in which situation this Kan extension is also T K -algebraic. We announce briefly the two ingredients of the construction the adjunction

j * ⊣ j * is T K -algebraic the morphism T M -algebraic f * • j * : B→ C is represented by a T K -algebraic morphism.
To capture those two situations, we introduce the following terminology.

Definition 9 (operadicity) A morphism f on the bicategory K is said to be T K -operadic when it is T K -algebraic and when its right adjoint f * in M is T M -algebraic.

As we will see later, whereas the notion of operadicity is purely algebraic, its meaning is inherently combinatorial. It indicates a kind of tree decomposition property. The second property is of an algebraic flavor. It must be understood as a way to say, in the monoidal setting, that the morphism colim -which intuitively computes the colimits of interest -commutes with the tensor product.

Definition 10 (algebraic cocompleteness) A C-cocomplete object C of the bicategory K is said to be T -algebraically C-cocomplete when C and C are both T -pseudoalgebras and when the morphisms y, y * and colim involved in the definition of C-cocompleteness are all T -algebraic morphisms.

Thanks to those to new notions, we can now state and prove the theorem which has motivated this paper.

Theorem 1

Let j : A → B be a T K -operadic morphism, C be a T -algebraically C-cocomplete object of the bicategory K and f :

A → C be a T -algebraic morphism. If the morphism f * • j * factories as B f * •j * -→ C = B g * -→ C y * -→ C
for every f , then the forgetful functor

U f : Ps-T K -Alg(B, C) → Ps-T K -Alg(A, C)
has a left adjoint computed with a left Kan extension

Lan j : Ps-T K -Alg(A, C) → Ps-T K -Alg(B, C).
Proof : The morphism f * • j * is the left Kan extension of f * along j * in M. According to Proposition 1, the morphism colim •g is the representative of f * • j * in K. But, according to Proposition 2, this representative is the left Kan extension Lan j f of f along j in K.

It now remains to check that Lan j f is T K -algebraic. As j is operadic, the morphism f * • j * is T M -algebraic. But the morphism y * is pseudomonic with respect to K so g is T K -algebraic. As colim is T K -algebraic, we conclude that Lan j f is T K -algebraic.

The functoriality of the construction is given by th pseudmonicity of y * .

For the particular case of the proarrow equipment Cat → Dist, we deduce from Theorem 1 that the free L 2 -model on a L 1 -model A along a morphism j : L 1 → L 2 is computed as the coend

Lan j A = m∈L 1 L 2 (f m, n) ⊗ A(m)
Since we discussed at length the conclusion of the theorem, we will directly comment the two hypotheses of operadicity and algebraic completeness.

A combinatorial hypothesis: operadicity

We have claimed that the notion of operadicity is of a combinatorial nature. Let us explain this intuition by making it more explicit in the proarrow equipment Cat -→ Dist. In that situation, a T -algebraic morphism j : L 1 -→ L 2 operadic when the canonical morphism

p∈T (L 1 ) L 1 (m, [p]) ⊗ T (L 2 )(T j(p), n) -→ L 2 (jm, [n]) (10)
is an isomorphism in the category Ens, for every object m of the category L 1 , and every object n of the category T (L 2 ).

The definition is purely algebraic, but its meaning is inherently combinatorial: Operadicity = tree decomposition property.

To understand why, it is worth expanding the definition of operadic functor in the particular case of linear theories -keeping in mind that an object m of the linear theory L 1 is a natural number, and an object n of the category

T (L 2 ) is a finite sequence (n 1 , • • • , n k ) of natural numbers.
Because the underlying category is the category Ens, the canonical morphism ( 10) is a function, whose domain is the set

p 1 ∈L 1 • • • p k ∈L 1 L 1 (m, p 1 + • • • + p k ) × L 2 (p 1 , n 1 ) × • • • × L 2 (p k , n k ) of pairs (g, h 1 , • • • , h k ) consisting of a morphism g : m - → p 1 + • • • + p k
in the linear theory L 1 , and a family of k morphisms

h i : p i - → n i (1 ≤ i ≤ k)
in the linear theory L 2 . These pairs are considered modulo the smallest equivalence relation ∼ satisfying:

(g, h 1 • j(h ′ 1 ), • • • , h k • j(h ′ k )) ∼ ((h ′ 1 ⊗ • • • ⊗ h ′ k ) • g, h 1 , • • • , h k ) whenever h ′ i : p i - → q i (1 ≤ i ≤ k)
is a family of morphisms in the linear theory L 1 . Now, the function ( 10) transports every such family (g,

h 1 , • • • , h k ) to the morphism (h 1 ⊗ • • • ⊗ h k ) • j(g) : m - → p 1 + • • • + p k - → n 1 + • • • + n k
of the linear theory L 2 . By operadic, we mean that the function ( 10) is a bijection, for every natural number m and sequence of natural numbers

(n 1 , • • • , n k ).
This should be seen as a tree decomposition property, which states that every morphism

h : m - → n 1 + • • • + n k
in the linear theory L 2 decomposes uniquely as a morphism

j(g) : m - → p 1 + • • • + p k followed by a morphism h 1 ⊗ • • • ⊗ h k : p 1 + • • • + p k - → n 1 + • • • + n k
modulo the equivalence relation ∼ defined above. The terminology "operadic" is justified by the fact the property holds for any map of operads j : L 1 -→ L 2 between operads -that is, linear theories L 1 and L 2 generated by an equational theory on operations m -→ 1 with m inputs and one output. In that case, the morphism g may be taken the identity, with m = p 1 + • • • + p k , and each morphism h i describing a particular "component" with n i roots of the morphism h understood as "forest" of operations with n 1 + • • • + n k roots.

=

In particular, the linear functor N -→ ∆ mentioned in ( 9) is operadic: this explains why the formula (1) for the tensor algebra, obtained by taking left Kan extensions along j, computes indeed the free monoid object in k-Mod, and defines a left adjoint to the forgetful functor U : k-Alg -→ k-Mod. Another example is provided by the linear functor from the linear theory of k-algebras to the linear theory of Lie algebras, which induces the notion of enveloping k-algebra. Another less immediate example is comonoids to bimonoids in PROPs.

To conclude, we give the counter-example of the inclusion functor from the trivial linear theory N to the theory of bimonoids. This explains the observation mentioned in the introduction that there exists no free k-bialgebra in general.

The case of algebraic theories and projective sketches. For algebraic theories and projective sketches, it is known for a long time that the computation of a free L 2 -model on a L 1 -model is always possible. It seems then that the hypothesis of operadicity is always valid. Let us take a look at the meaning of this condition in the cartesian setting. The canonical morphism 10 becomes

p 1 ∈L 1 • • • p k ∈L 1 L 1 (m, p 1 × • • • × p k ) × L 2 (jp 1 , n 1 ) × • • • × L 2 (jp k , n k ). (11)
We can use the surjective pairing to deduce that

L 1 (m, p 1 × • • • × p k ) ∼ = L 1 (m, p 1 ) × • • • × L 1 (m, p k ).
Then, using the Yoneda lemma, we have that the object 11 is isomorphic to

L 2 (jm, n 1 ) × • • • × L 2 (jm, n k )
and we conclude by using again surjective pairing. But there is a more abstract reasoning that explains this property for the two pseudomonads. Those two pseudomonads are KZ-doctrines. It follows that the T -algebraic structure a : T L 1 → L 1 of a category L 1 is the right adjoint to the unit η L 1 of the pseudomonad T . This means that there is an isomorphism

L 1 (m, a(p)) ∼ = T L 1 (η A (m), p)
We can use this adjunction to show that the operadicity property always holds

p∈T (L 1 ) L 1 (m, a(p)) ⊗ T (L 2 )(T j(p), n) ∼ = p∈T (L 1 ) T L 1 (η L 1 (m), p) ⊗ T (L 2 )(T j(p), n) (a ⊢ η L 1 ) ) ∼ = T L 2 (T j(η L 1 (m))n) (Yoneda lemma) ∼ = T L 2 (η L 2 (jm), n) (naturality of η) ∼ = L 2 (jm, [n]) (a ⊢ η L 1 )
We deduce a well-known result, which is a corollary of Theorem 1, and which states that the computation of free models on Ens is always possible for algebraic theories and projective sketches.

Corollary 1 Let j be a algebraic morphism between two algebraic theories or two projective sketches L 1 et L 2 . Every L 1 -model A induces a free L 2 -model Lan j A, we have a bijection Model(L 2 , Ens)(Lan j A, B)

∼ = - -→ Model(L 1 , Ens)(A, U j B)
More precisely, the left Kan extension Lan j A defines a left adjoint functor to the forgetful functor:

Lan j ⊣ U j : Model(L 1 , Ens) → Model(L 2 , Ens).

An algebraic hypothesis: algebraic cocompleteness

A category is called cocomplete when it has all small colimits. When the category is moreover monoidal, one generally requires that "colimits commute to the tensor product".

ϕ ⊗ ψ : c → c 1 c 2 C(c, c 1 ⊗ c 2 ) × ϕ(c 1 ) × ψ(c 2 ).
The factorization system on Cat ensures that the diagram that corresponds to this presheaf is the functor ⊗ • (F × G) indexed by I × J as shown by the following diagram

I × J final G G F ×G A A S S S S S S S S S S S S S S S S S Elt(ϕ) × Elt(ψ) discrete fibration final G G Elt(ϕ ⊗ ψ) discrete fibration C × C ⊗ G G C
We see in that case that if the colimits of F and G exist and commute with the tensor product, then the colimit of ⊗ • (F × G) exists and commutes with the tensor product. It is then natural to consider the category C of presheaves having a colimit that commutes with the tensor product. The only remaining thing to do in concrete cases is to check that the distributor on which we compute the representative factories through this category.

Computing the free monoid

We will now apply our theory to the computation of the free monoid. As depicted in the introduction, this amounts to compute the free ∆-model on a N-model in the linear theories setting. Those two theories N and ∆ come from operads and the inclusion morphism j N : N → ∆ is obviously operadic. It just remains to check that the required colimits in the category C exist and commute with the tensor product (in what follows, we will say monoidal colimit).

A simplified verification

Let C be an object of a monoidal category C and F : N → C be the associated trivial model. Recall that F n is equal to c ⊗n for all n of N. We have to check that the distributor F * •j * N factories through the category C of presheaves having a monoidal colimit. At first glance, we must check that the presheaf

F * • j * N (-, n) = k∈N ∆(Jk, n) × C(-, c ⊗k )
has a monoidal colimit, and this for all n ∈ ∆. Since the distributor F * • j * N is strong monoidal, it suffices to study the case n = 1 because then, the isomorphism ,1) ⊗n ensures that the other presheaves also have a monoidal colimit. But the object 1 is terminal in the category ∆, so the expression of F * • j * N (-, 1) can be drastically simplified into

F * • j * N (-, n) ∼ = F * • j * N (-
F * • j * N (-, 1) = colim k∈N C(-, c ⊗k ).
We have already seen that a diagram corresponding to a presheaf of that kind is precisely the functor F : N → C. We thus deduce the following proposition.

Proposition 4 Let C be a monoidal category and c be an object of C. If the colimit of diagram

F : N → C n → c ⊗n
exists and commutes with the tensor product, then it defines the free monoidal on C.

We have just proved, with a detour by proarrow equipments, that Equations ( 1) and ( 8) define respectively the free algebra in k-Mod and the free monoid in Ens. But now that we have the general theory, we can deal with more complicated and less known cases.

A diagram as a colimit of diagrams

Various constructions of free monoids [START_REF] Dubuc | Free monoids[END_REF][START_REF] Vallette | Free monoid in monoidal abelian categories[END_REF][START_REF] Lack | Note on the construction of free monoids[END_REF] shed light on the need to know that the colimit of a diagram is monoidal, by using general properties of commutation of some kinds of colimits. For example, a common hypothesis is that the considered category has all coequalizers and that they commute with the tensor product. We present here a mean to ensure that a diagram has a monoidal colimit by showing that this is the monoidal colimit of diagrams which have themselves a monoidal colimit. More formally, we have the following property.

Proposition 5 Let C be a monoidal category and C be the category of diagram on C having a monoidal colimit. Suppose that all diagrams, whose base is the category J, have a monoidal colimit. Then the category C is closed for diagrams indexed on J.

C(ϕ, ι • y(d)) ∼ = [J, C](ι • F, ι • y(d) • t J ) ϕ is the colimit of ι • F ∼ = [J, C](F, y(d) • t J ) ι is pseudomonic ∼ = [J, C](colim •F, d • t J ) colim ⊢ y ∼ = C(c, d) c is the colimit of colim •F 4.4
The Vallette/Lack's construction Bruno Vallette notices in his paper [START_REF] Vallette | Free monoid in monoidal abelian categories[END_REF] that the hypothesis that coequalizers commute with the tensor product is often too strong. He points out various situations where only reflexive coequalizers commute with the tensor product. Let us recall briefly what is a reflexive coequalizer.

Definition 11 (reflexive coequalizer) A pair of morphisms f, g : A → B is said to be reflexive when their exists a morphism i : B → A that makes the following diagram commute

A f G G g G G B i Ñ Ñ
A reflexive coequalizer is a coequalizer computed on a reflexive pair.

When the category of consideration is equipped with finite coproducts, it is well-known that every pair of morphisms can be replaced by a reflexive pair computing the same equalizer.

Proposition 8 Let C be a category equipped with finite coproducts (noted ⊕) and let f, g : A → B be a pair of morphisms of C. The coequalizer of f and g exists if and only if the reflexive coequalizer of the reflexive pair

A ⊕ B f ⊕id B G G g⊕id B G G B i 2

ẽxists

(i 2 is the second injection).

We see appearing a variation on Proposition 6, where the computation of A n using coequalizers is replaced by a computation using reflexive coequalizers. This is precisely the result stated by Bruno Vallette and then reformulated by Steve Lack.

Proposition 9 Let C be a monoidal category equipped with finite coproducts and such that • all reflexive coequalizers exist and commute with the tensor product,

• all sequential colimits (in the sense of colimits of a diagram indexed on the category N) exist and commute with the tensor product, Then, the free monoid on an object A : N → C exists and is computed by the image in 1 of the left Kan extension of the pointed object A ⊕ 1 along j ∆ .

Proof : We go back to the proof of Propositions 6 and 7 with the same notations. The only difference is that we can not use general coequalizers to compute the colimit of the diagram A n and equivalently of the diagram D n . Anyway, we will use Proposition 8 to compute the colimit of D n using reflexive coequalizers. The first computed coequalizer can be replaced by the monoidal reflexive coequalizer as follows

A n-1 i1 G G (A ⊕ A 2 ) ⊗ A n-2 (A(d 2 1 )⊕id A 2 )⊗id A n-2 G G (A(d 2 2 )⊕id A 2 )⊗id A n-2 G G A 2 ⊗ A n-2 i2⊗id A n-2 x x e1⊗A n-2 G G A 2 ⊗ A n-2
And so on. . . The rest of the proof is unchanged.

Computing free commutative comonoid

We are also interested in computing free commutative comonoid, that correspond to free exponential modality in linear logic. But we first have to solve a problem: since the beginning, we deal with monoids but never with comonoids. This difficulty is not deep as a comonoid in the monoidal category C is nothing but a monoid in the opposite category C op equipped with the induced tensor product.

One can thus recycle what has been done for the computation of the free monoid by just dualizing every diagram and computing limits instead of colimits. Let us state, for example, a direct corollary of Proposition 6.

Corollary 2 Let C be a monoidal category such that

• the equalizers exist and commute with the tensor product,

• the sequential limits (in the sense of limits of diagrams indexed on the category N) exist and commute with the tensor product.

Then the free comonoid on the copointed object A : ∆ face op → C exists and is computed by the image on 1 of the right Kan extension of A on j ∆ op :

Ran j ∆ op A(1) = lim n∈∆ face op A(n).

As we are interested in the commutative case, we have to move from PROs to PROPs. We have already mentioned that the category Bij of finite sets and bijections is the trivial symmetric theory. In the same way, the category FinSet of finite sets and set theoretic functions is the symmetric theory for commutative monoids.

The injection of Bij into FinSet is operadic, so we directly obtain the following proposition.

Proposition 10 Let C be an algebraically cocomplete symmetric monoidal category. The free commutative comonoid ! • e A on an object A of C seen as a symmetric monoidal functor from Bij to C is computed by the following formula

! • e A = lim n∈Bij A n
For example, this construction applies to the category Rel of sets and relations. We get, as expected, the construction of the set of finite multisets on a set

! • e A = n∈Bij A n / ∼ Bij .
We find back the usual exponential of the relational model of linear logic.

Extension to the enriched case 6 Conclusion and future works

We have presented a framework to compute free models of various kind of theories, from algebraic, linear, symmetric theories to projective sketches. This construction, which is based on proarrow equipment, lies on two fundamental properties: operadicity and algebraic cocompleteness.

In future work, we will see how our formulation of algebraic cocompleteness enables to compute the free commutative comonoid in very poorly equipped categories such as coherence spaces or the category of Conway games.

1 1 . 7

 117 The tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 A basic problem in algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Lawvere theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Linear theories (PROs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Monoidal Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Symmetric theories (PROPs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.8 T -algebraic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.9 Algebraic distributors at work . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Proarrow equipments 12 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Yoneda situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Pseudomonads in proarrow equipments . . . . . . . . . . . . . . . . . . . . . . 15 1 Introduction

  [n] -→ [n]. And the symmetric theory of bimonoids B is defined as the symmetric monoidal category whose morphisms m -→ n are (a) the m × n matrices with integer coefficients in Z, or equivalently, (b) the homomorphisms Z[m] -→ Z[n] between free commutative monoids -where Z[m] is the free commutative monoid on m elements.

  from the enriched 2-category of enriched categories to the enriched bicategory of enriched distributors. • K = Map B and M = B for every bicategory B, where Map B is the full sub-bicategory of B restricted to the morphisms which have a right adjoint. (-) * is then an inclusion functor, • K = Ens and M = Rel where (-) * sends every function f : A → B to its graph f * af * b ssi f a = b.

Definition 7 (

 7 C-cocompleteness) An object C of K is said to be C-cocomplete iff there exists a Yoneda situation y : C → C, that has a left adjoint colim ⊣ y : C → C.

  Fine... But what does this mean? We clarify the concept by reformulating it in the language of proarrow equipment, namely as a property of algebraic cocompleteness. Let us look at what it means in the proarrow equipment Cat → Dist. Informally speaking, C-algebraic cocompleteness = colimits computed on a diagram indexed by an element of C commute with the T -algebraic structure. Let us specialize once more and look at the case of linear theories. The category C is T -algebraically C-cocomplete when the class of indices F is closed under the cartesian product of categories and when the functor colim is monoidal. Given two functors F : I → C and G : J → C and their corresponding presheaves ϕ and ψ with I, J ∈ F, one can form the Day's tensor product of ϕ and ψ as described by the coend formula

Proof:

  Consider a diagram F : J → C indexed on J. We have to show that the colimit of this diagram exists in C. Let colim : C → C be a functor that associates to every diagram of C its colimit in C and ι : C → C the injection from C to the category of presheaves on C. By hypothesis on J, the diagram colim •F has a monoidal colimit in C, noted c. Since C is the free completion by colimit of C, the diagram ι • F has a colimit ϕ in C. It thus suffices then to show that the presheaf ϕ is represented by c. Let d be an object C, t J : J → 1 and y : C → C be the Yoneda embedding restricted to C. The following chain of equivalences

This work has been supported by the ANR Project INVAL "Invariants algébriques des systèmes informatiques".

shows that c is a representative of ϕ. We then deduce that the colimit of ϕ exists and is monoidal, that is to say ϕ ∈ C. This proposition sheds a new light on the constructions of Eduardo Dubuc and of Bruno Vallette (refined by Steve Lack).

Th Dubuc's construction

In a 1974's paper, Eduardo Dubuc [START_REF] Dubuc | Free monoids[END_REF] gives a construction of the free monoid on a pointed object in a monoidal category C. This construction involves a transfinite process that is not of our concern. Nevertheless, we can give here an interpretation of the ideas developed by Dubuc.

First, we have to introduce the linear theory of pointed object. This is nothing but the category ∆ face of augmented simplicial sets and injective maps. This is a sub-category of ∆ in which we only keep non-degenerate morphisms. We note d n i : n -1 → n the face morphism corresponding to d n i : {1, . . . , n -1} → {1, . . . , i -1, i + 1, . . . , n}. Thus, on can see every pointed object p : 1 → A of C as a strict functor A : ∆ face → C which associate to n the object A ⊗n and

Again, this linear theory is described by an operad and the morphism j ∆ : ∆ face → ∆ is a map of operad. As for Proposition 4, to obtain the free monoid on a pointed object A : ∆ face → C, it is enough to check that the colimit of A exists and commutes with the tensor product. This is the case in the following general case.

Proposition 6 Let C be a monoidal category such that • all coequalizers exist and commute with the tensor product,

• all sequential colimits (in the sense of colimits of a diagram indexed on the category N) exist and commute with the tensor product, Then, the free monoid on a pointed object A : ∆ face → C exists and is computed by the image in 1 of the left Kan extension of A along j ∆ :

Let ∆ face (n) be the full sub-category of ∆ face of integers below n. First, we establish that the colimit of the restriction of the diagram A to ∆ face (n)

Remark that this diagram has the same colimit than the diagram:

Remark also that this diagram is computed by coequalizing the morphisms two-by-two. Indeed, because the coequalizers are monoidal, the colimit of the two arrows on the top is given by the coequalizer e 1 : A ⊗2 → A 2 of the two faces d 2 1 and d 2 2 , tensored with A ⊗n-2

A ⊗n-1

Then, we compute the coequalizer of the morphism

And we start again until we obtain the monoidal colimit of the diagram D n , or equivalently of the diagram A n . Thus, A n lives in C for all n. We conclude by noticing that A is the sequential colimit of the functor

and by using Proposition 5.

We can refine the theorem when the category C is equipped with finite coproducts. It is then possible to compute the free pointed object on an object. This pointed object is simply defined by

Proposition 7 Let C be a monoidal category equipped with finite coproducts and such that • all coequalizers exist and commute with the tensor product,

• all sequential colimits (in the sense of colimits of a diagram indexed on the category N) exist and commute with the tensor product, Then, the free monoid on an object A : N → C exists and is computed by the image in 1 of the left Kan extension of the pointed object A ⊕ 1 along j ∆ .

Proof : There exists an inclusion functor j ′ N : N → ∆ face which factories j N through j ∆ . Since Kan extensions do compose, we can deduce that the free monoid on the free pointed object on an object A is also the free monoid on this object. According to Proposition 6, it is enough to check that A ⊕ 1 is the free pointed object on A.

Let f : A → P be a morphism of C to a pointed object (P, p). We define the morphism of pointed object f ⊕p : (A⊕1, i 2 ) → (P, p). This morphism fits and is unique by universality of the coproduct.

Remark that when the category C has finite coproducts which do not commute with the tensor product, the construction of the free pointed object is not obtained by left Kan extension in Cat. Indeed, using the left Kan extension, we get a functor Lan j N A which is not monoidal as indicated by the diagram bellow, Lan j N A(2) = 1 ⊕ A ⊕ A ⊕ A 2 ≇ (A ⊕ 1) ⊗ (A ⊕ 1) = Lan j N A(1) ⊗ Lan j N A(1).

Thus, Proposition 7 combines a "pedestrian construction" of the monoidal left Kan extension on j ′ N : N → ∆ face with the computation of the left Kan extension on j ∆ : ∆ face → ∆ that is known, for abstract reasons, to be monoidal.