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Medea

In Greek mythology, Medea (Μήδεια; lit. 'planner, schemer') is the daughter of King Aeëtes of Colchis. In the myth of Jason and the Argonauts, she aids Jason in his search for the Golden Fleece. She later marries him, but eventually kills his children and his other bride. Medea is known in most stories as a sorceress and is often depicted as a priestess of the goddess Hecate. She first appears in Hesiod's Theogony around 700 BCE, but is best known from Euripides's tragedy Medea and Apollonius of Rhodes's epic Argonautica.

As a daughter of King Aeëtes she is a mythical granddaughter of the sun god Helios and a niece of Circe, an enchantress goddess. Her mother might have been Idyia. She plays the archetypal role of helper-maiden, aiding Jason in his search for the Golden Fleece. In said quest, she uses her magic to save his life and kills her brother to allow Jason to escape. Once he finishes his quest, she abandons her native home of Colchis and flees westwards with Jason, where they eventually settle in Corinth and marry.

Euripides's 5th-century BCE tragedy Medea depicts the ending of her union with Jason, when after ten years of marriage, Jason intends to abandon her to wed King Creon's daughter Creusa. Jason argues that their sons will have a better future in the city if he marries the King's daughter and his sons stay in the same city with him. He claims the sacred vows he has with Medea can be broken because she is a foreigner, which makes their marriage illegitimate. Medea is exiled from Corinth by King Creon, and is offered refuge in Athens by King Aegeus after she offers to help him get an heir with her magic.

In revenge against Jason, Medea murders her own sons and Jason's new bride, King Creon's daughter, with a poisoned crown and robes, so that Jason will be without heir and legacy for the rest of his life. What happens afterwards varies according to several accounts. Herodotus in his Histories mentions that she ended up leaving Athens and settling in the Iranian plateau among the Aryans, who subsequently changed their name to the Medes.

Genealogy and divinity
Medea is a direct descendant of the sun god Helios (son of the Titan Hyperion) through her father King Aeëtes of Colchis. According to Hesiod (Theogony 956–962), Helios and the Oceanid Perseis produced two children, Circe and Aeëtes. Aeëtes then married the Oceanid Idyia and Medea was their child. From here, Medea's family tree becomes a little more complicated and disputed. By some accounts, Aeëtes and Idyia only had two daughters, Medea and Chalciope (or Chalkiope). There was one son, Absyrtus (or Apsyrtus), who was the son of Aeëtes through Asterodea. This would make him a half-brother to Medea herself. According to others, Idyia gave birth to Medea and Apsyrtus while Asterodea gave birth to Chalciope. Even with the two differing accounts, it is known that Medea has a sister and a brother.

As she becomes older, Medea marries Jason and together they have children. The number and names of their children are questioned by scholars. Depending on the account, it is two to fourteen children. In his play, Medea, Euripides mentions two unnamed sons. According to other accounts, her children were "Mermerus, Pheres or Thessalus, Alcimenes and Tisander, and according to others, she had seven sons and seven daughters, while others mention only two children, Medus (some call him Polyxenus) and Eriopis, or one son Argos." No matter the number of children, Medea eventually leaves Jason in Corinth, and marries the King of Athens (Aegeus) and bears him a son. While with him, it is questioned if that was when she had her son Medeius, who goes on to become the ancestor of the Medes by conquering their lands.

Understanding Medea's genealogy helps define her divinity. By some accounts, like the Argonautica, she is depicted as a young, mortal woman who is directly influenced by the Greek gods Hera and Aphrodite. While she possesses magical abilities, she is still a mortal with divine ancestry. Other accounts, like Euripides's play Medea, focus on her mortality. Hesiod's Theogony places her marriage to Jason on the list of marriages between mortals and divine, suggesting that she is predominantly divine. She also has connections with Hecate, the goddess of magic, which could be one of the main sources from which she draws her magical ties. Although distinct from the Titan known as Perses, who is known for fathering Hecate, the goddess of witchcraft, Diodorus Siculus in his Bibliotheca historica made Perses of Colchis the father of Hecate by an unknown mother; Perses' brother Aeëtes then married Hecate and had Medea and Circe by her.

Jason and Medea

Medea is first introduced in Greek Mythology after Jason came from Iolcus to Colchis in an attempt to claim his inheritance and throne by retrieving the Golden Fleece. In the most complete surviving account, the Argonautica of Apollonius of Rhodes, Hera convinced Aphrodite or Eros to cast a spell on Medea so that she would fall in love with Jason and promise her skills to help him. She does promise her skills, but only if he agreed to marry her. Jason agreed, knowing Medea and her powers would help him in the long run. In a familiar mythic motif, Jason is promised the Golden Fleece through Aeëtes, but only if Jason could complete a list of tasks. The first harrowing task was ploughing a field with fire-breathing oxen that Jason had to yoke himself. To aid him in this, Medea gave him an unguent to anoint himself and his weapons, to protect them from the bulls' fiery breath. After ploughing the field, Jason had to sow the teeth of a dragon. This task seemed to be somewhat simplistic but Medea forewarned him that the teeth would spring into soldiers. To combat this, she told him to throw a rock into the crowd to cause confusion among the soldiers. The soldiers, now confused, would then begin to attack and kill each other instead of Jason. For the last task, Aeëtes assigned Jason to fight and kill the sleepless dragon that guarded the fleece. Medea aided Jason in this task by putting the beast to sleep with her narcotic herbs. Once the dragon was asleep, Jason then took the fleece and sailed away with Medea as promised. Medea distracted her father as they fled by killing her brother Absyrtus.

In some versions, Medea was said to have dismembered her brother's body and scattered his parts on an island, knowing her father would stop to retrieve them for proper burial; in other versions, it was Absyrtus himself who pursued them and was killed by Jason. However, in the Argonautica, Medea and Jason stopped on her aunt Circe's island so that she could be cleansed after murdering her brother, relieving her of blame for the deed. This is one of the times we see Medea use her powers. During the fight, Atalanta, someone helping Jason in his quest, was seriously wounded. Medea was able to use her powers to heal the wound. On the way back to Thessaly, Medea prophesied that Euphemus, the helmsman of Jason's ship, the Argo, would one day rule over all of Libya. Pindar alleges that this came true through Battus, saying that he was a distant descendant of Euphemus (by 17 generations).

After the prophecy, the Argo reached the island of Crete, guarded by the bronze man, Talos (Talus). Talos had one vein which went from his neck to his ankle, bound shut by a single bronze nail. According to Apollodorus, Talos was slain either when Medea drove him mad with drugs, deceived him that she would make him immortal by removing the nail, or was killed by Poeas's arrow. In the Argonautica, Medea hypnotized him from the Argo, driving him mad so that he dislodged the nail, ichor flowed from the wound, and he bled to death. After Talos died, the Argo landed.

At some point while in Thessaly, Medea and the Nereid Thetis argued over which one was the most beautiful. They appointed the Cretan Idomeneus as the judge, who declared Thetis to be the most beautiful. In her anger, Medea called all Cretans liars, and cursed them to never say the truth.

Jason, celebrating his return with the Golden Fleece, noted that his father Aeson was too aged and infirm to participate in the celebrations. Medea understood the impact this had on Jason and was able to invigorate him by withdrawing the blood from Aeson's body, infused it with certain herbs, and returning it to his veins. The daughters of King Pelias saw this and asked Medea to perform the same service on their father. Medea agreed.

However, the service was never performed. Hera, who was angry at Pelias, conspired to make Jason fall in love with Medea, who, Hera hoped, would kill Pelias. Hera's plan worked, and the pair fell in love with each other. When they returned to Iolcus, Pelias refused to give up his throne to Jason. Jason had been promised the throne in turn for the Golden Fleece. So, Medea conspired to have Pelias's own daughters kill him. She demonstrated her powers to them by showing her cutting up an old ram and putting the pieces in stew. Once the pieces were in, Medea added some magic herbs and stirred the concoction, a young ram suddenly jumping out of the stew. Excited at the sight, the girls cut their father into pieces and threw him into a pot. Unfortunately, the King never came to life. Having killed Pelias, Jason and Medea fled to Corinth.

While in Corinth, the couple were married and lived together for 10 years. They had between one and fourteen children depending on the source. The known children are sons Alcimenes, Thessalus, Tisander, Mermeros and Pheres, Medus, and Argos, and a daughter, Eriopis. As well as having children, a myth states that Medea ended a famine in Corinth by sacrificing to Demeter and the nymphs. Zeus then desired her, but she declined his advances in order not to incur Hera's wrath. As a reward, Hera offered to make her children immortal.

Various Myth endings
In Corinth, Jason abandoned Medea for the king's daughter, Glauce. Before the fifth century BCE, there seem to have been two variants of the myth's conclusion. According to the poet Eumelus, to whom the fragmentary epic Korinthiaka is usually attributed, Medea killed her children by accident. She buried them alive in the Temple of Hera, believing this would make them immortal. The poet Creophylus, however, blamed their murders on the citizens of Corinth.

According to Euripides's version, Medea took her revenge by sending Glauce a dress and golden coronet, covered in poison. This resulted in the deaths of both the princess and the king, Creon, when he went to save his daughter. Medea then continued her revenge, murdering two of her children herself and refusing to allow Jason to hold the bodies. Afterward, she left Corinth and flew to Athens in a golden chariot driven by dragons sent by her grandfather, Helios, god of the sun.

Although Jason in Euripides calls Medea most hateful to gods and men, the fact that the chariot is given to her by Helios indicates that she has the Gods on her side. As Bernard Knox points out, Medea's last scene parallels that of a number of indisputably divine beings in other plays by Euripides. Just like these gods, Medea "interrupts and puts a stop to the violent action of the human being on the lower level" and "justifies her savage revenge on the grounds that she has been treated with disrespect and mockery" so that she "takes measures and gives orders for the burial of the dead, prophesies the future," and "announces the foundation of a cult." This deliberate murder of her children by Medea appears to be Euripides's invention, although some scholars believe Neophron created this alternate tradition. Her filicide would go on to become the standard for later writers. Pausanias, writing in the late 2nd century CE, records five different versions of what happened to Medea's children after reporting that he has seen a monument for them while traveling in Corinth. Fleeing from Jason, Medea made her way to Thebes, where she healed Heracles (the former Argonaut) from the curse of Hera (that led him to slay his sons).

After the murder of her children, Medea fled to Athens, where she met and married Aegeus. They had one son, Medus. Another version from Hesiod makes Medus the son of Jason. Her domestic bliss was once again shattered by the arrival of Aegeus's long-lost son, Theseus. Determined to preserve her own son's inheritance, Medea convinced her husband that Theseus was an imposter, making him a threat and that he needed to be disposed of. To do this, Medea was planning on poisoning him as she previously had other victims. As Medea handed Theseus a cup of poison, Aegeus recognized the young man's sword as his own, which he had left behind many years previously for his newborn son as soon as he came of age. Knocking the cup from Medea's hand, Aegeus embraced Theseus as his own.

Medea returned to Colchis and found that Aeëtes had been deposed by his brother Perses, which prompted her to kill her uncle and restore the kingdom to her father. Herodotus reports another version, in which Medea and her son Medus fled from Athens, on her flying chariot. They landed in the Iranian plateau and lived among the Aryans, who then changed their name to the Medes.

Recounting the many variations of Medea's story, the 1st century BCE historian Diodorus Siculus wrote, "Speaking generally, it is because of the desire of the tragic poets for the marvelous that so varied and inconsistent an account of Medea has been given out."

Personae of Medea

In Euripides's play Medea, she is a woman scorned, rejected by her husband Jason and revenge seeking. Deborah Boedeker writes about different images and symbolism Euripides used in his play to evoke responses from his original Athenian audience. The Nurse, one of the characters, gives descriptions of Medea in the prologue, highlighting comparisons to great forces of nature and different animals. There are also many nautical references throughout the play either used by other characters when describing Medea or by Medea herself. By including these references, Boedeker argues that these comparisons were used to create connections to the type of woman Medea was. She holds great power (referred to by the comparisons to forces of nature), she relies on her basic animal-like instincts and emotions (connections to different animals like bulls and lions), and it draws the audience back to her original myth of Jason's quest for the Golden Fleece and the sea voyage taken by Jason, Medea, and the Argonauts.

Emma Griffiths also adds to the analysis of Medea's character in Euripides's play by discussing the male/female dichotomy created by Euripides. Medea does not fit into the mold of a "normal woman" according to Athenian philosophy. She is depicted as having great intelligence and skill, traits typically viewed as masculine by Euripides's original audience. On the other hand, she uses her intelligence to manipulate the men around her. This manipulation would have been a negative female trait to the Athenian audience. Griffiths also acknowledges the paradox of the methods Medea uses to kill. She poisons the princess, which would have been seen as a feminine way of murder, yet kills her children in cold blood, which is seen as more masculine. Medea is also shown as a 'normal' Athenian mother by having a dialogue about her children and showing a strong maternal love and connection to them. Yet at the end of the play, she is able to kill her children as part of her revenge. It is through these opposites that Euripides creates a complicated character for his protagonist.

Marianne McDonald argues that "Medea's anger turns to violent action, which can make her into a symbol of freedom, and emblem for the colonized turning the tables on the colonizer. Euripides, more than all other tragedians, has predicted many of the horrors that occur in the modern world, showing both the glory and the monstrosity of the oppressed turned oppressor."

Although not the first depiction of Medea, the Argonautica by Apollonios Rhodios gives a fuller description of events that lead up to Euripides's play, mainly surrounding Jason's quest for the Golden Fleece. In this literary work, Medea is presented not as a powerful woman seeking justice, but as a young woman who is desperately in love with Jason. So much in love that she decides to defy her father and kill her brother in order to help him. James J. Clauss writes about this Medea, attempting to unearth another version of this character for scholarship and discussion. He looks into different passages in the original text to define the meaning and draw connection to the different feelings Medea was going through. He argues the feelings of Medea's initial love for Jason, the shame she feels for loving him and for going against her family, and final agreement to help Jason in his quest.

Multiple scholars have discussed Medea's use as a "helper maiden" to Jason's quest. A helper maid is typically a young woman who helps on a hero's quest, usually out of love. Instead of being the center of the story, like she is in Euripides's Medea, this version of Medea is reduced to a supporting role. Her main purpose is to help the hero with his quest. Jason would never have been successful on his quest without Medea's help, something that is pointed out and referenced many times in ancient texts and contemporary scholarly work.

Other, non-literary traditions guided the vase-painters, and a localized, chthonic presence of Medea was propitiated with unrecorded emotional overtones at Corinth, at the sanctuary devoted to her slain children, or locally venerated elsewhere as a foundress of cities.

Medea as Divine Agent
A mortal with divine ancestry –Medea is the granddaughter of Helios– she seems to kill without consequence, suggesting that she acts as a divine force and is immune from vengeance that typically comes to mortals who commit injustice. After Medea kills her brother Apsyrtos, she, Jason, and the others stop at Circe's island to be absolved of the crime. When they return to Thessaly, Medea convinces Pelias's daughters to kill their father with trickery. After this, they are an accessory to a murder, so they must leave. They go to Korinth, where Jason proceeds to court and marry Glauke, a princess of Korinth, where Jason loses Hera's favor and gains Medea's wrath. King Kreon tells Medea that she has twenty-four hours to leave, and if she doesn't she'll be killed. Jason comes to scold Medea and she in turn accuses him of denying his oath to the gods. Despite her previous crimes, Medea seems to have the approval of the gods. Next Medea resolves her time to kill Glauke with a potion that causes her to catch fire. Her father Kreon dies also when he in grief hugs his daughter and dies from the same poison. Medea proceeds to kill her and Jason's children as well, and before Jason can stop her, she is escorted away on a flying chariot sent by her grandfather, Helios. Later, Medea marries King Aegeus and the two produce a son named Medus. When Theseus returns in an attempt to prove he is Aegeus's son, Medea recognizes that he will be the heir to the throne rather than Medus, and convinces Aegeus to poison Theseus's drink. Aegeus realizes Theseus is his son and throws the drink away. Medea is forced to leave with Medus, where they return to Kolkhis where Medus eventually claims the title of king. Medea sits in an odd position where though she kills many people, she never seems to face any divine consequence from it. Perhaps her actions are in keeping with the favor of the gods. True or not, Medea is framed as a curiously nuanced figure, something rare for women in Greek Mythology.

Modern references
In his book The Happiness Hypothesis, social psychologist Jonathan Haidt quotes Medea talking about her struggle between her love to Jason and obligation to her father, as an example for "The divided Self" and the conflict between Id and Superego:

"I am dragged along by a strange new force. Desire and reason are pulling in different directions. I see the right way and approve it, but follow the wrong.

Written sources (in chronological order)

 * Hesiod, Theogony 1000-2
 * Herodotus, Histories I.2 and VII.62i
 * Pindar, Pythian Odes, IV
 * Neophron, Medea (fragments from the play)
 * Euripides, Medea
 * Apollonius Rhodius, Argonautica
 * Plautus, Pseudolus 869–871
 * Diodorus Siculus, Bibliotheca Historica
 * Ovid
 * Heroides XII
 * Metamorphoses VII, 1–450
 * Tristia iii.9

 * Seneca the Younger: Medea (tragedy)
 * Hyginus, Fabulae 21–26
 * Gaius Valerius Flaccus Argonautica (epic)
 * Pseudo-Apollodorus Bibliotheca I, 23–28

title: Medea


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Icarus (disambiguation)

Icarus is a character in Greek mythology.

Icarus or Ikarus may also refer to:

People

 * Roger Squires (1932–2023), crossword compiler who has used the pseudonym Icarus
 * Icarus (wrestler) (born 1982), wrestler with the Chikara organization

Places

 * Icarus (island), in the Aegean Sea
 * Ikaros (Failaka Island), in the Persian Gulf

Aircraft

 * Hartman Ikarus, a 1950s British human-powered ornithopter
 * Icarus I, Icarus II and Icarus V, series of rigid-wing hang gliders designed by Taras Kiceniuk, Jr.
 * Ikarus C42, a microlight aircraft
 * Ikarus IK-2, the most notable indigenous Yugoslav aircraft of the 1930s and World War II era
 * Icarus, a prototype space rescue vehicle tested by Aleksandr Serebrov

Aviation businesses and organizations

 * Ikarus Drachen Tomas Pellicci, a German hang glider manufacturer
 * Icaro Air, an Ecuadorian airline
 * Icarus School, a military aviation academy
 * Icarus, a Greek airline that preceded Olympic Airlines
 * Ikarus, a Yugoslav aircraft manufacturer later renamed Ikarbus

Other uses in aviation

 * Coupe Icare ("Icarus Cup"), a flight festival

Other businesses and organizations

 * Icarus Project, a mental health organization
 * Icarus Theatre Collective, a British theatre company
 * Ikarus (Hungarian company), a Hungarian bus manufacturer
 * Ikarbus, a Serbian bus manufacturer formerly known as Ikarus
 * Icarus FC, an association football kit and sportswear brand

Film and television

 * Icarus (Planet of the Apes), a spacecraft in the Planet of the Apes franchise
 * Icarus Base, a place in the series Stargate Universe
 * Icarus, an orbital superweapon in the James Bond film Die Another Day
 * Icarus I, and Icarus II, spacecraft in Sunshine
 * Icarus (2010 film) (also The Killing Machine), an action film
 * Icarus (2017 film), a documentary film by Bryan Fogel
 * "Icarus" (Scandal), an episode of the TV series Scandal
 * "Icarus", the eleventh episode of Smallville, Season 10
 * "Icarus", the seventh episode of Law & Order: Criminal Intent

Fictional characters

 * Icarus (comics), a Marvel Comics character
 * Icarus, a mecha in Chōjin Sentai Jetman
 * Icarus, a Deus Ex character
 * Icarus, a character in Hercules: The Animated Series
 * Icarus, a character in Little Nemo: Adventures in Slumberland
 * Ikaros, a character in Heaven's Lost Property
 * Icarus, a primary character based on the Greek myth in the Cirque du Soleil show Varekai

Music

 * Icaros, shamanic medicine songs
 * Icarus (band), a band from London, England

Albums

 * Icarus (The Forms album), 2003
 * Icarus (Chicosci album), 2004
 * Icarus (Paul Winter Consort album), 1972
 * Icarus, 2011 EP by Periphery
 * Icaro, album by Renato Zero
 * Icarus (Cryalot EP), 2022

Songs

 * "Icarus" (Greeeen song), 2015
 * "Icarus" (Madeon song), 2012
 * "Icarus" (R3hab song), 2016
 * "Icarus", a song by Santigold on the album Top Ranking
 * "Icarus", a song by Jason Webley on the album Only Just Beginning
 * "Ikarus", a song by Unheilig on Phosphor
 * "Icarus", a song by Bastille on the album Bad Blood
 * "Icarus", a song by Ralph Towner on the album Diary
 * "Icarus - Borne on Wings of Steel", a song by Kansas on the album Masque
 * "Icarus", a song by JJ Project on the EP Verse 2, 2017
 * "Icarus", a song by Eden on the album vertigo
 * "Icarus", a song by Emma Blackery from the 2018 album Villains
 * "Icarus", a song by Starset on the album Horizons, 2021

Other media

 * Icarus (magazine), a Trinity College student publication
 * Icarus (sculpture), 1973 by Michael Ayrton

Software and video games

 * Icarus (video game), a sci-fi survival video game developed by RocketWerkz
 * Ikarus (typography software), a type design and production software developed by URW foundry
 * Ikarus (Scheme implementation), a free software optimizing incremental compiler for R6RS Scheme
 * Icarus Verilog, an implementation of the Verilog hardware description language
 * BioShock Infinite, codenamed "Project Icarus" during development
 * Ikarus (chess), a computer chess program

Space

 * 1566 Icarus, an asteroid
 * ICARUS experiment, a neutrino detector at Gran Sasso National Laboratory, Italy
 * Icarus (crater), on the Moon
 * Icarus (journal), a planetary science journal
 * Icarus (star), most distant individual star detected (as of April 2018)
 * IKAROS, an interplanetary uncrewed solar sail spacecraft
 * Project Icarus (interstellar), a design study of an interstellar spacecraft based on Project Daedalus
 * Project Icarus (photography) (2009), a project at the Massachusetts Institute of Technology (MIT)

Other uses in science and technology

 * Icarus (plant), a fern genus
 * ICARUS Initiative, an international project studying the migration routes of flying animals
 * IKZF1, a protein in humans that is encoded by the IKZF genes of the Ikaros family zinc finger group

Other uses

 * HMS Icarus, the name of several ships
 * USCGC Icarus (WPC-110), a patrol boat of the United States Coast Guard
 * Ikarus, a restaurant in Hangar-7 in Salzburg, Austria
 * Icarus affair, an incident involving the 1867 murder of two British sailors in Nagasaki
 * Operation Ikarus, a World War II German plan to invade Iceland

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Icarus (disambiguation)


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Limit (category theory)

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.

Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.

Definition
Limits and colimits in a category $$C$$ are defined by means of diagrams in $$C$$. Formally, a diagram of shape $$J$$ in $$C$$ is a functor from $$J$$ to $$C$$:
 * $$F:J\to C.$$

The category $$J$$ is thought of as an index category, and the diagram $$F$$ is thought of as indexing a collection of objects and morphisms in $$C$$ patterned on $$J$$.

One is most often interested in the case where the category $$J$$ is a small or even finite category. A diagram is said to be small or finite whenever $$J$$ is.

Limits
Let $$F : J \to C$$ be a diagram of shape $$J$$ in a category $$C$$. A cone to $$F$$ is an object $$N$$ of $$C$$ together with a family $$\psi_X:N\to F(X)$$ of morphisms indexed by the objects $$X$$ of $$J$$, such that for every morphism $$f: X \to Y$$ in $$J$$, we have $$F(f)\circ\psi_X=\psi_Y$$.

A limit of the diagram $$F:J\to C$$ is a cone $$(L, \phi)$$ to $$F$$ such that for every cone $$(N, \psi)$$ to $$F$$ there exists a unique morphism $$u:N\to L$$ such that $$\phi_X\circ u=\psi_X$$ for all $$X$$ in $$J$$. One says that the cone $$(N, \psi)$$ factors through the cone $$(L, \phi)$$ with the unique factorization $$u$$. The morphism $$u$$ is sometimes called the mediating morphism.

Limits are also referred to as universal cones, since they are characterized by a universal property (see below for more information). As with every universal property, the above definition describes a balanced state of generality: The limit object $$L$$ has to be general enough to allow any cone to factor through it; on the other hand, $$L$$ has to be sufficiently specific, so that only one such factorization is possible for every cone.

Limits may also be characterized as terminal objects in the category of cones to F.

It is possible that a diagram does not have a limit at all. However, if a diagram does have a limit then this limit is essentially unique: it is unique up to a unique isomorphism. For this reason one often speaks of the limit of F.

Colimits
The dual notions of limits and cones are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here:

A co-cone of a diagram $$F:J\to C$$ is an object $$N$$ of $$C$$ together with a family of morphisms
 * $$\psi_X:F(X) \to N$$

for every object $$X$$ of $$J$$, such that for every morphism $$f:X\to Y$$ in $$J$$, we have $$\psi_Y\circ F(f)=\psi_X$$.

A colimit of a diagram $$F:J\to C$$ is a co-cone $$(L, \phi)$$ of $$F$$ such that for any other co-cone $$(N, \psi)$$ of $$F$$ there exists a unique morphism $$u:L\to N$$ such that $$u\circ \phi_X = \psi_X$$ for all $$X$$ in $$J$$.

Colimits are also referred to as universal co-cones. They can be characterized as initial objects in the category of co-cones from $$F$$.

As with limits, if a diagram $$F$$ has a colimit then this colimit is unique up to a unique isomorphism.

Variations
Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams. The definitions are the same (note that in definitions above we never needed to use composition of morphisms in $$J$$). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) directed graph $$G$$. If we let $$J$$ be the free category generated by $$G$$, there is a universal diagram $$F:J\to C$$ whose image contains $$G$$. The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection of objects and morphisms.

Weak limit and weak colimits are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped.

Limits
The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consider the limit (L, φ) of a diagram F : J → C.
 * Terminal objects. If J is the empty category there is only one diagram of shape J: the empty one (similar to the empty function in set theory). A cone to the empty diagram is essentially just an object of C. The limit of F is any object that is uniquely factored through by every other object. This is just the definition of a terminal object.
 * Products. If J is a discrete category then a diagram F is essentially nothing but a family of objects of C, indexed by J. The limit L of F is called the product of these objects. The cone φ consists of a family of morphisms φX : L → F(X) called the projections of the product. In the category of sets, for instance, the products are given by Cartesian products and the projections are just the natural projections onto the various factors.
 * Powers. A special case of a product is when the diagram F is a constant functor to an object X of C. The limit of this diagram is called the Jth power of X and denoted XJ.
 * Equalizers. If J is a category with two objects and two parallel morphisms from one object to the other, then a diagram of shape J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms.
 * Kernels. A kernel is a special case of an equalizer where one of the morphisms is a zero morphism.
 * Pullbacks. Let F be a diagram that picks out three objects X, Y, and Z in C, where the only non-identity morphisms are f : X → Z and g : Y → Z. The limit L of F is called a pullback or a fiber product. It can nicely be visualized as a commutative square:
 * Inverse limits. Let J be a directed set (considered as a small category by adding arrows i → j if and only if i ≥ j) and let F : Jop → C be a diagram. The limit of F is called an inverse limit or projective limit.
 * If J = 1, the category with a single object and morphism, then a diagram of shape J is essentially just an object X of C. A cone to an object X is just a morphism with codomain X. A morphism f : Y → X is a limit of the diagram X if and only if f is an isomorphism. More generally, if J is any category with an initial object i, then any diagram of shape J has a limit, namely any object isomorphic to F(i). Such an isomorphism uniquely determines a universal cone to F.
 * Topological limits. Limits of functions are a special case of limits of filters, which are related to categorical limits as follows. Given a topological space X, denote by F the set of filters on X, x ∈ X a point, V(x) ∈ F the neighborhood filter of x, A ∈ F a particular filter and $$F_{x,A}=\{G\in F \mid V(x)\cup A \subset G\} $$ the set of filters finer than A and that converge to x. The filters F are given a small and thin category structure by adding an arrow A → B if and only if A ⊆ B. The injection $$I_{x,A}:F_{x,A}\to F$$ becomes a functor and the following equivalence holds :

 * x is a topological limit of A if and only if A is a categorical limit of $$I_{x,A}$$

Colimits
Examples of colimits are given by the dual versions of the examples above:
 * Initial objects are colimits of empty diagrams.
 * Coproducts are colimits of diagrams indexed by discrete categories.
 * Copowers are colimits of constant diagrams from discrete categories.
 * Coequalizers are colimits of a parallel pair of morphisms.
 * Cokernels are coequalizers of a morphism and a parallel zero morphism.
 * Pushouts are colimits of a pair of morphisms with common domain.
 * Direct limits are colimits of diagrams indexed by directed sets.

Existence of limits
A given diagram F : J → C may or may not have a limit (or colimit) in C. Indeed, there may not even be a cone to F, let alone a universal cone.

A category C is said to have limits of shape J if every diagram of shape J has a limit in C. Specifically, a category C is said to A complete category is a category that has all small limits (i.e. all limits of shape J for every small category J).
 * have products if it has limits of shape J for every small discrete category J (it need not have large products),
 * have equalizers if it has limits of shape $$\bullet\rightrightarrows\bullet$$ (i.e. every parallel pair of morphisms has an equalizer),
 * have pullbacks if it has limits of shape $$\bullet\rightarrow\bullet\leftarrow\bullet$$ (i.e. every pair of morphisms with common codomain has a pullback).

One can also make the dual definitions. A category has colimits of shape J if every diagram of shape J has a colimit in C. A cocomplete category is one that has all small colimits.

The existence theorem for limits states that if a category C has equalizers and all products indexed by the classes Ob(J) and Hom(J), then C has all limits of shape J. In this case, the limit of a diagram F : J → C can be constructed as the equalizer of the two morphisms
 * $$s,t : \prod_{i\in\operatorname{Ob}(J)}F(i) \rightrightarrows \prod_{f\in\operatorname{Hom}(J)} F(\operatorname{cod}(f))$$

given (in component form) by
 * $$\begin{align}

s &= \bigl( F(f)\circ\pi_{\operatorname{dom}(f)}\bigr)_{f\in\operatorname{Hom}(J)} \\ t &= \bigl( \pi_{\operatorname{cod}(f)}\bigr)_{f\in\operatorname{Hom}(J)}. \end{align}$$ There is a dual existence theorem for colimits in terms of coequalizers and coproducts. Both of these theorems give sufficient and necessary conditions for the existence of all (co)limits of shape J.

Universal property
Limits and colimits are important special cases of universal constructions.

Let C be a category and let J be a small index category. The functor category CJ may be thought of as the category of all diagrams of shape J in C. The diagonal functor
 * $$\Delta : \mathcal C \to \mathcal C^{\mathcal J}$$

is the functor that maps each object N in C to the constant functor Δ(N) : J → C to N. That is, Δ(N)(X) = N for each object X in J and Δ(N)(f) = idN for each morphism f in J.

Given a diagram F: J → C (thought of as an object in CJ), a natural transformation ψ : Δ(N) → F (which is just a morphism in the category CJ) is the same thing as a cone from N to F. To see this, first note that Δ(N)(X) = N for all X implies that the components of ψ are morphisms ψX : N → F(X), which all share the domain N. Moreover, the requirement that the cone's diagrams commute is true simply because this ψ is a natural transformation. (Dually, a natural transformation ψ : F → Δ(N) is the same thing as a co-cone from F to N.)

Therefore, the definitions of limits and colimits can then be restated in the form:
 * A limit of F is a universal morphism from Δ to F.
 * A colimit of F is a universal morphism from F to Δ.

Adjunctions
Like all universal constructions, the formation of limits and colimits is functorial in nature. In other words, if every diagram of shape J has a limit in C (for J small) there exists a limit functor
 * $$\lim : \mathcal{C}^\mathcal{J} \to \mathcal{C}$$

which assigns each diagram its limit and each natural transformation η : F → G the unique morphism lim η : lim F → lim G commuting with the corresponding universal cones. This functor is right adjoint to the diagonal functor Δ : C → CJ. This adjunction gives a bijection between the set of all morphisms from N to lim F and the set of all cones from N to F
 * $$\operatorname{Hom}(N,\lim F) \cong \operatorname{Cone}(N,F)$$

which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected (and nonempty) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ. This fails if J is not connected. For example, if J is a discrete category, the components of the unit are the diagonal morphisms δ : N → NJ.

Dually, if every diagram of shape J has a colimit in C (for J small) there exists a colimit functor
 * $$\operatorname{colim} : \mathcal{C}^\mathcal{J} \to \mathcal{C}$$

which assigns each diagram its colimit. This functor is left adjoint to the diagonal functor Δ : C → CJ, and one has a natural isomorphism
 * $$\operatorname{Hom}(\operatorname{colim}F,N) \cong \operatorname{Cocone}(F,N).$$

The unit of this adjunction is the universal cocone from F to colim F. If J is connected (and nonempty) then the counit is an isomorphism, so that colim is a left inverse of Δ.

Note that both the limit and the colimit functors are covariant functors.

As representations of functors
One can use Hom functors to relate limits and colimits in a category C to limits in Set, the category of sets. This follows, in part, from the fact the covariant Hom functor Hom(N, –) : C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits.

If a diagram F : J → C has a limit in C, denoted by lim F, there is a canonical isomorphism
 * $$\operatorname{Hom}(N,\lim F)\cong \lim\operatorname{Hom}(N,F-)$$

which is natural in the variable N. Here the functor Hom(N, F–) is the composition of the Hom functor Hom(N, –) with F. This isomorphism is the unique one which respects the limiting cones.

One can use the above relationship to define the limit of F in C. The first step is to observe that the limit of the functor Hom(N, F–) can be identified with the set of all cones from N to F:
 * $$\lim\operatorname{Hom}(N,F-) = \operatorname{Cone}(N,F).$$

The limiting cone is given by the family of maps πX : Cone(N, F) → Hom(N, FX) where $\pi$X(ψ) = ψX. If one is given an object L of C together with a natural isomorphism Φ : Hom(L, –) → Cone(–, F), the object L will be a limit of F with the limiting cone given by ΦL(idL). In fancy language, this amounts to saying that a limit of F is a representation of the functor Cone(–, F) : C → Set.

Dually, if a diagram F : J → C has a colimit in C, denoted colim F, there is a unique canonical isomorphism
 * $$\operatorname{Hom}(\operatorname{colim} F, N)\cong\lim\operatorname{Hom}(F-,N)$$

which is natural in the variable N and respects the colimiting cones. Identifying the limit of Hom(F–, N) with the set Cocone(F, N), this relationship can be used to define the colimit of the diagram F as a representation of the functor Cocone(F, –).

Interchange of limits and colimits of sets
Let I be a finite category and J be a small filtered category. For any bifunctor

 * $$F : I\times J \to \mathbf{Set},$$

there is a natural isomorphism

 * $$\operatorname{colim}\limits_J \lim_I F(i, j) \rightarrow \lim_I\operatorname{colim}\limits_J F(i, j).$$

In words, filtered colimits in Set commute with finite limits. It also holds that small colimits commute with small limits.

Functors and limits
If F : J → C is a diagram in C and G : C → D is a functor then by composition (recall that a diagram is just a functor) one obtains a diagram GF : J → D. A natural question is then:
 * “How are the limits of GF related to those of F?”

Preservation of limits
A functor G : C → D induces a map from Cone(F) to Cone(GF): if Ψ is a cone from N to F then GΨ is a cone from GN to GF. The functor G is said to preserve the limits of F if (GL, Gφ) is a limit of GF whenever (L, φ) is a limit of F. (Note that if the limit of F does not exist, then G vacuously preserves the limits of F.)

A functor G is said to preserve all limits of shape J if it preserves the limits of all diagrams F : J → C. For example, one can say that G preserves products, equalizers, pullbacks, etc. A continuous functor is one that preserves all small limits.

One can make analogous definitions for colimits. For instance, a functor G preserves the colimits of F if G(L, φ) is a colimit of GF whenever (L, φ) is a colimit of F. A cocontinuous functor is one that preserves all small colimits.

If C is a complete category, then, by the above existence theorem for limits, a functor G : C → D is continuous if and only if it preserves (small) products and equalizers. Dually, G is cocontinuous if and only if it preserves (small) coproducts and coequalizers.

An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous. Since adjoint functors exist in abundance, this gives numerous examples of continuous and cocontinuous functors.

For a given diagram F : J → C and functor G : C → D, if both F and GF have specified limits there is a unique canonical morphism
 * $$\tau_F : G \lim F \to \lim GF$$

which respects the corresponding limit cones. The functor G preserves the limits of F if and only if this map is an isomorphism. If the categories C and D have all limits of shape J then lim is a functor and the morphisms τF form the components of a natural transformation
 * $$\tau:G \lim \to \lim G^J.$$

The functor G preserves all limits of shape J if and only if τ is a natural isomorphism. In this sense, the functor G can be said to commute with limits (up to a canonical natural isomorphism).

Preservation of limits and colimits is a concept that only applies to covariant functors. For contravariant functors the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.

Lifting of limits
A functor G : C → D is said to lift limits for a diagram F : J → C if whenever (L, φ) is a limit of GF there exists a limit (L′, φ′) of F such that G(L′, φ′) = (L, φ). A functor G lifts limits of shape J if it lifts limits for all diagrams of shape J. One can therefore talk about lifting products, equalizers, pullbacks, etc. Finally, one says that G lifts limits if it lifts all limits. There are dual definitions for the lifting of colimits.

A functor G lifts limits uniquely for a diagram F if there is a unique preimage cone (L′, φ′) such that (L′, φ′) is a limit of F and G(L′, φ′) = (L, φ). One can show that G lifts limits uniquely if and only if it lifts limits and is amnestic.

Lifting of limits is clearly related to preservation of limits. If G lifts limits for a diagram F and GF has a limit, then F also has a limit and G preserves the limits of F. It follows that: The dual statements for colimits are equally valid.
 * If G lifts limits of all shape J and D has all limits of shape J, then C also has all limits of shape J and G preserves these limits.
 * If G lifts all small limits and D is complete, then C is also complete and G is continuous.

Creation and reflection of limits
Let F : J → C be a diagram. A functor G : C → D is said to Dually, one can define creation and reflection of colimits.
 * create limits for F if whenever (L, φ) is a limit of GF there exists a unique cone (L′, φ′) to F such that G(L′, φ′) = (L, φ), and furthermore, this cone is a limit of F.
 * reflect limits for F if each cone to F whose image under G is a limit of GF is already a limit of F.

The following statements are easily seen to be equivalent: There are examples of functors which lift limits uniquely but neither create nor reflect them.
 * The functor G creates limits.
 * The functor G lifts limits uniquely and reflects limits.

Examples

 * Every representable functor C → Set preserves limits (but not necessarily colimits). In particular, for any object A of C, this is true of the covariant Hom functor Hom(A,–) : C → Set.
 * The forgetful functor U : Grp → Set creates (and preserves) all small limits and filtered colimits; however, U does not preserve coproducts. This situation is typical of algebraic forgetful functors.
 * The free functor F : Set → Grp (which assigns to every set S the free group over S) is left adjoint to forgetful functor U and is, therefore, cocontinuous. This explains why the free product of two free groups G and H is the free group generated by the disjoint union of the generators of G and H.
 * The inclusion functor Ab → Grp creates limits but does not preserve coproducts (the coproduct of two abelian groups being the direct sum).
 * The forgetful functor Top → Set lifts limits and colimits uniquely but creates neither.
 * Let Metc be the category of metric spaces with continuous functions for morphisms. The forgetful functor Metc → Set lifts finite limits but does not lift them uniquely.

A note on terminology
Older terminology referred to limits as "inverse limits" or "projective limits", and to colimits as "direct limits" or "inductive limits". This has been the source of a lot of confusion.

There are several ways to remember the modern terminology. First of all, are types of colimits, whereas are types of limits. Second, the prefix "co" implies "first variable of the $$\operatorname{Hom}$$". Terms like "cohomology" and "cofibration" all have a slightly stronger association with the first variable, i.e., the contravariant variable, of the $$\operatorname{Hom}$$ bifunctor.
 * cokernels,
 * coproducts,
 * coequalizers, and
 * codomains
 * kernels,
 * products
 * equalizers, and
 * domains

title: Limit (category theory)