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Introduction to Category Theory/Products and Coproducts

Product
One of the features of category theory is how it unifies very disparate phenomena. The product construction comprises for example both the cartesian product of sets, and the greatest lower bound (meet) of posets and lattices. Here's a definition:

Definition
The product of objects A and B is an object P together with the morphisms $$\pi_1: P \to A$$ and $$\pi_2: P \to B$$ that satisfy the following universal property: The universal property has two parts:
 * If $$X$$ is any object in the category, and $$f_1: X \to A$$ and $$f_2: X \to B$$ are any morphisms, then there exists a unique morphism $$g: X \to P$$ such that $$f_1 = \pi_1 \circ g$$ and $$f_2 = \pi_2 \circ g$$. In other words there is one and only one morphism g such that both triangles in the diagram commute.
 * 1) There exists a morphism g.
 * 2) The morphism g is unique.

The object P alone isn't a product. The product consists of the object P and two morphisms $$\pi_1\;$$ and $$\pi_2\;$$. Often these morphisms are clear from the context, and we simply say let $$A \times B$$ be the product of A and B without specifying the morphisms.

Not every category has products for all pairs of objects (i.e. 'has all products'). For example in the category with 3 objects and 2 arrows (+identity arrows) shown at right, the product of A and C is the object A together with morphisms $$1_A\;$$ and $$A \to C$$. But this category has no product $$A \times B$$ (why?).

The following proposition allows us to speak of the product of A and B.

Proposition. The product of any two objects is unique up to isomorphism.

Proof. If $$(P,\pi_1,\pi_2)\;$$ and $$(X,f_1,f_2)\;$$ are both products of $$A$$ and $$B$$, then by the first part of universal property there are morphisms $$g:X \to P$$ and $$h:P \to X$$ such that
 * $$\begin{cases}

f_1 = \pi_1 \circ g \\ f_2 = \pi_2 \circ g \end{cases}$$ and $$\begin{cases} \pi_1 = f_1 \circ h \\ \pi_2 = f_2 \circ h \end{cases}$$ Therefore
 * $$\begin{cases}

f_1 \circ 1_X = f_1 = \pi_1 \circ g = f_1 \circ h \circ g \\ f_2 \circ 1_X = f_2 = \pi_2 \circ g = f_2 \circ h \circ g \end{cases}$$ and by the uniqueness part of the universal property of X, we must have $$1_X = h \circ g$$. By similar argument $$1_P = g \circ h$$ by the universal property of P. Thus g and h are inverses of each other, and P and X are isomorphic. $$\Box$$

Here is an alternative definition, where we leverage our earlier work on terminal objects, and define the product as a terminal object in a spinoff category. If you compare the two closely, you'll see that they're really not very different at all, just different packaging of the same ideas.

We say that a category 'has products' when there is a product for every pair of objects in the category. It is possible for a category to have products for some but not all pairs of objects. We say that a category has 'designated products' when there is some specific technique for manufacturing the product from the objects of the pair. Cartesian products of sets, discussed in the examples section below, provide an example of designated products.

Examples
Proposition. The cartesian product of non-empty sets is a product in the category of sets.

Proof. Let A and B be any non-empty sets, $$A \times B$$ their cartesian product set, and $$\pi_1, \pi_2\;$$ the canonical coordinate projection maps. We must prove that $$A \times B$$ satisfies the universal property of product.

Let X be any set and let $$f_1:X \to A$$ and $$f_2:X \to B$$ be any functions. We construct the function $$g:X \to A \times B$$ by setting $$g(x) = (f_1(x), f_2(x))\;$$ for every element x in X. This satisfies the first part of universal property:
 * $$(\pi_1 \circ g)(x) = \pi_1(g(x)) = \pi_1(\ (f_1(x), f_2(x))\ ) = f_1(x)$$ and
 * $$(\pi_2 \circ g)(x) = \pi_2(g(x)) = \pi_2(\ (f_1(x), f_2(x))\ ) = f_2(x)$$.

We must prove that if $$h:X \to A \times B$$ is another function with $$\pi_1 \circ h = f_1$$ and $$\pi_2 \circ h = f_2$$, then g=h. Let x be any element in X, then h(x) is an element (a,b) in AxB, for some a in A and b in B. An easy computation shows
 * $$f_1(x) = (\pi_1 \circ h)(x) = \pi_1(h(x)) = \pi_1((a,b)) = a$$ and
 * $$f_2(x) = (\pi_2 \circ h)(x) = \pi_2(h(x)) = \pi_2((a,b)) = b$$.

Therefore $$h(x) = (a,b) = (f_1(x), f_2(x)) = g(x)\;$$ for every x in X, and h=g is the unique function required by the second part of universal property. $$\Box$$

Note that the proposition remains true if we consider empty sets. The product $$A \times \emptyset$$ is $$\emptyset$$. Exercise: Check that the universal property holds - there are not very many sets with a map $$f_2$$ to the empty set.

More on this example below

Proposition. Intersection of subsets is a product in the powerset poset.

Proof. Let X be any set, and let A and B be any subsets of X. We must prove that the intersection $$A \cap B$$ has the universal property.

Let C be any subset of X such that there are arrows $$C \to A$$ and $$C \to B$$. By the definition of powerset poset, there's an arrow from C to A if and only if C is a subset of A, in other words if every element of C is an element of A. So every element of C is an element of both A and B. The definition of intersection says that an element x belongs to the intersection $$A \cap B$$ if and only if x belongs to both A and B. Thus C is a subset of $$A \cap B$$ and there is an arrow $$C \to A \cap B$$. Uniqueness and composition property follow from the fact that in any poset there is at most one arrow between any two objects. $$\Box$$

Proposition. Least common multiple lcm is a product in the divisibility poset of positive integers.

Proof. Let a and b be positive integers. We must prove that the least common multiple of a and b, denoted lcm(a,b), has the universal property.

Let c be any positive integer divisible by both a and b. There are integers q and r (quotient and remainder), such that $$c = q*\text{lcm}(a,b) + r\;$$ with $$q>0\;$$ and $$0 \le r < lcm(a,b)$$. (Why is q positive and not zero?) Since c and lcm(a,b) are both divisible by a and b, also r must be divisible by them. But r is smaller than lcm(a,b), so r must be zero or otherwise lcm isn't the least common multiple. Thus c = q*lcm(a,b) and c is divisible by lcm(a,b). By the definition of poset there is at most one arrow between any two objects, so the uniqueness part is trivial. $$\Box$$

Proposition. Greatest lower bound in a poset, when it exists for two elements, is a (actually the unique, not just up to isomorphism) product of those two elements. This provides lots of examples where products might exist for some, all, or no pairs of elements in a category,

Proof. Exercise.

Products and Elements
The unique map $$g\,$$ above, from $$C\,$$ to $$A\times B$$, is often notated as $$\,$$, using the standard ordered pair notation from set-theory. This is motivated by the special case in the category $$\mathcal Set$$ where $$C=1$$. In this case, $$f\,$$ and $$g\,$$ will be categorical 'elements', and effectively the same thing as set-theoretical elements, since a function from a one-element set picks out a unique element from its codomain. Now $$\,$$ will be a categorical element of $$A\times B$$, that is, essentially the same thing as a set-theoretical element of $$A\times B$$, that is, an ordered pair whose first member is from $$A\,$$ and second from $$B\,$$.

This point is often dramatized by using variable letters such as $$x,y\,$$ when discussing categorical elements.

Something we get from the categorical formulation of cartesian products and ordered pairs is explicit recognition of the fact that it doesn't matter what actual construction we use to produce the ordered pairs - all that matters is the existence of the projections and the universal property, which can be seen as saying that the ordered pair must contain just enough information for the projections to extract the first components, and no more.

You can see an interactive demonstration of universality here. Clicking the "generate a product and demonstrate its universality" button will generate a product in the category of finite sets. It will also generate another object, and show the universal arrow from this to the product.

This demonstration also shows that it doesn't matter what construction we use to produce the product. Clicking the "generate a product to show that many products exist" button will generate a product that is not a set of ordered pairs, and will show its projections.

Alternative Definition
Here is an alternative definition of product, which procedes by way of a spinoff category. If it seems too hard to understand, feel free to ignore it (comments about whether it is useful would be welcome).

For any pair of objects $$A, B\,$$, define the 'product-maker' category $$\Pi(A,B)\,$$ as follows: If you draw the commutative diagram for these equations, you'll see that, except for some of the names, it looks just like the diagram for the product, and the reason for this will become evident shortly, if you haven't seen it already. But first, you might want to work throught the details of why the product-maker for $$A,B\,$$ is a category (or maybe not, if the point seems obvious enough).
 * the objects of $$\Pi(A,B)\,$$ are triples of the form $$(C, f, g)\,$$ where $$f:C\rightarrow A$$ and $$g:C\rightarrow B$$
 * the arrows of $$\operatorname{Hom}((C, f, g),(C',f',g'))$$ are the arrows $$h\,$$ of $$\operatorname{Hom}(C,C')$$ such that:
 * $$f = f'\circ h$$
 * $$g=g'\circ h$$

And so, the point of all this is the following: $$A\times B$$ is the/a terminal object of $$\Pi(A,B)\,$$.

A benefit of defining products in this way is that we don't have to bother to prove that they are unique up to isomorphism, since this is already known as a property of terminal objects. On the other hand we had to prove that the product-maker is a category.

You can see an interactive demonstration of terminality here. Clicking the "Terminality" button will generate a product object in the category of finite sets, and another object in the product-maker category, and show the arrow from that to the product.

Associativity of Product
Proposition. Product is associative, in other words if A, B, and C are objects in a category with products, then $$(A \times B) \times C \cong A \times (B \times C)$$. Technically, this is a 'weak' rather than 'strict' form of associativity, since we require only isomorphism, not full identity. In category theory, the weak versions of notions such as associativity are usually more important that the strict ones (although these are not always irrelevant).

Proof. The following diagram shows products and their projection morphisms. We will prove that there exist morphisms g and i that are inverses of each other. By the universal property of the product of B and C there exists morphism f such that
 * $$\begin{cases}

q_1 \circ f = p_2 \circ r_1\\ q_2 \circ f = r_2 \end{cases}$$ By the universal property of the product of A and BxC there exists morphism g such that
 * $$\begin{cases}

s_1 \circ g = p_1 \circ r_1\\ s_2 \circ g = f \end{cases}$$ Similarly there exists morphisms h and i such that
 * $$\begin{cases}

p_1 \circ h = s_1\\ p_2 \circ h = q_1 \circ s_2 \end{cases}$$
 * $$\begin{cases}

r_1 \circ i = h\\ r_2 \circ i = q_2 \circ s_2 \end{cases}$$ A straightforward calculation shows
 * $$\begin{cases}

q_1 \circ s_2 \circ g \circ i = q_1 \circ f \circ i = p_2 \circ r_1 \circ i = p_2 \circ h = q_1 \circ s_2\\ q_2 \circ s_2 \circ g \circ i = q_2 \circ f \circ i = r_2 \circ i = q_2 \circ s_2 \end{cases}$$ By the uniqueness part of the universal property of BxC it follows that $$s_2 \circ g \circ i = s_2$$.
 * $$s_1 \circ g \circ i = p_1 \circ r_1 \circ i = p_1 \circ h = s_1$$

and by the universal property of Ax(BxC) we conclude that $$g \circ i = 1_{A \times (B \times C)}$$. Similarly reader can verify that $$i \circ g = 1_{(A \times B) \times C}$$.

Corollary 1. Cartesian product of sets is associative, that is if A, B, and C are sets, then $$(A \times B) \times C \cong A \times (B \times C)$$.

Proof. We have showed that cartesian product of sets is product in the category of sets, thus claim follows by abstract nonsense.

Corollary 2. Intersection of subsets is associative, that is if A, B, and C are subsets of some set X, then $$(A \cap B) \cap C = A \cap (B \cap C)$$.

Proof. We have showed that intersection of subsets is product in the powerset poset, and we also know that only isomorphisms in a poset are identity morphisms, thus claim follows by abstract nonsense. (But this associativity is strict rather than weak, which our present level of abstract nonsense, at least, does not suffice to prove).

Corollary 3. Least common multiple is associative, that is if a, b, and c are positive integers, then lcm(lcm(a,b), c) = lcm(a, lcm(b,c)). (strictly or weakly?)

Proof. We have showed that least common multiple is product in the divisibility poset, thus claim follows by abstract nonsense.

Remark. A preorder is a poset without the antisymmetry property. We can define a divisibility preorder that also has negative numbers. In this case n is divisible by -n for every integer n. It follows that n is isomorphic to -n, and product in preorder is associative up to isomorphism, in other words lcm(lcm(a,b), c) = ± lcm(a, lcm(b,c)).

Corollary 4. In a poset, products are strictly associative.

Proof. Exercise

Ternary and n-ary Products
What we did for pairs of objects can be carried through for triples, n-tuples or even 0-tuples. Suppose we have objects $$A_1, A_2, A_3\,$$, along with an object $$P\,$$ and arrows $$\pi_1,\pi_2, \pi_3\,$$ from $$P\,$$ to $$A_1, A_2, A_3\,$$, respectively. $$(P, \pi_1,\pi_2, \pi_3)\,$$ constitutes a ternary product of $$A_1, A_2, A_3\,$$ iff, for any object $$C\,$$ and arrows $$f_1, f_2, f_3\,$$ from $$C\,$$ to $$A_1, A_2, A_3\,$$, respectively, there is a unique arrow $$h = :C\rightarrow P\,$$such that $$f_i = \pi_i\circ h$$ for $$i=1,2,3\,$$ [DIAGRAM TO COME]

We can do likewise for any $$n>1$$, although the result, when written out, is one of those things that is harder to read than to compose for yourself, so is relegated to the Hints and Details section. What about the case where $$n=1$$, 'unary products'? If you write out the appropriate diagram (try to do this before looking), you can see that any object $$A\,$$ will serve as its own unary product, with the identity morphism $$1_A\,$$ as the projection. So unlike the other cases, unary products exist for all objects in any category.

So, finally, what if $$n=0\,$$? You might think about this and see what you come up with before reading on.

By a natural extension of the preceeding definitions, a 0-ary product in a category $$\mathcal C\,$$ will be some object $$P\,$$ of $$\mathcal C$$ such that for any object $$C\,$$ of $$\mathcal C\,$$, there will be a unique arrow $$h:C\rightarrow P$$ (no involved objects, therefore no projections, and no composition of $$h\,$$ with any projections). This is just a terminal object of $$\mathcal C$$

We are not aware of any notion of n-ary product for negative n.

Finite Products
If a category has products for all non-negative integers n, we say that it has finite products. Happily, when true, this is relatively easy to prove, because all we have to do is show that the category has terminal objects and binary products: as noted above, a terminal object is a 0-ary product, and all categories have unary products (for each object). And n-ary products for $$n>2$$ can be obtained by iterating binary products. For example, $$(A\times B)\times C$$, with the associated projections, is a fine trinary product of $$A\,$$, $$B\,$$ and $$C\,$$. To formulate and prove this in generality is conceptually easy, but involves a fairly high level of notational masochism, which we won't indulge in.

A category with finite products is called a cartesian category, and by now you should be able to appreciate this essay

Coproduct
Any categorical notion has a 'dual', obtained by reversing the direction of the relevant arrows. If we do this with the product, we get the co-product, defined ab initio as follows:

Definition
[DIAGRAM TO COME] Object $$P\,$$ and morpisms $$\iota_1:A\rightarrow P$$, $$\iota_2:B\rightarrow P$$ are the/a co-product of $$A\,$$ and $$B\,$$ iff, for any object $$C\,$$ and arrows $$f_1:A\rightarrow C, f_2:B\rightarrow C$$, there is a unique arrow $$g:P\rightarrow C$$ such that: The coproduct object is usually written $$A+B$$, and the unique induced arrow $$g:P\rightarrow C$$ as $$[f,g]\,$$. The arrows $$\iota_1, \iota_2\,$$ to the coproduct object are called injections.
 * $$f_1= g \circ \iota_1$$
 * $$f_2= g \circ \iota_2$$

You can see an example of a coproduct and its injections in this interactive demonstration. Clicking the "generate a coproduct and demonstrate its universality" will generate an example of a coproduct, with its injections.

Just as the product can be defined as a terminal object in a spinoff category, so can the co-product be defined as an initial object in such a category.

The relevant spinoff category, which I'll notate as $$\coprod(A_1,A_2)$$ has as objects triples of the form $$(B, f_1, f_2)\,$$, where $$f_i:A_i\rightarrow B$$. And, for arrows, the arrows of $$\operatorname{Hom}((B, f_1, f_2), (C, g_1, g_2))$$ are those arrows $$h\,$$ of $$\operatorname{Hom}(B,C)$$ such that $$g_i=h\circ f_i$$ for $$i=1,2\,$$. Now $$A+B\,$$ is the/a initial object in this category.

Consequences
It follows from the above that a co-product is a product in the opposite category, which delivers many results with no additional work, such as: N-ary coproducts are denoted either $$\bigoplus_{1 \le i \le n}A_i$$ or $$\coprod_{1 \le i \le n}A_i$$.
 * co-products are unique up to isomorphism
 * co-products are associative up to isomorphism
 * if there are binary co-products for all pairs of objects, there are n-ary co-products for all $$n>0$$

What about 0-ary co-products? Clearly, to maintain duality, we need to define the 0-ary co-product as an initial object, and eveything then works out.

Examples

 * In a poset, a co-product is a least upper bound
 * A bounded lattice is a poset with finite products.
 * In the category of sets, the coproduct is the disjoint union, formed from two sets by processing the members of each differently, so that the results can't overlap, and then taking the ordinary union.  This coproduct is meaty enough to get more discussion below, and can be seen as the motivation for the use of the $$+\,$$ notation for co-products (which are,  unsurprisingly, also sometimes called sums).

Disjoint Unions
Suppose we have two sets $$A\,$$ and $$B\,$$, and we wish to form a coproduct of these. We can't just take their union, with the mappings $$x\mapsto x$$ from $$A\,$$ to $$A\cup B$$ and $$B\,$$ to $$A\cup B$$ because they might have some elements in common, which would make it possible to come up with functions $$f:A\rightarrow C$$ and $$g:B\rightarrow C$$ such that there can't be any $$[f|g]:A\cup B\rightarrow C$$ meeting the conditions for a co-product (Hint). But since we're doing category theory, where we usually regard isomorphic objects as equivalent, we can in this case just grab two sets $$A'\,$$ and $$B'\,$$ which don't overlap, and are isomorphic to $$A\,$$ and $$B\,$$, respectively. Their union $$A'\cup B'$$, will now provide a coproduct for $$A\,$$ and $$B\,$$.

Exercises:
 * what are the injections for the coproduct we've just made
 * show that if we added some additional elements to $$A'\cup B'$$, we'd no longer satisfy the conditions for a coproduct,

Now to demonstrate the $$\mathcal Set$$ indeed has coproducts, we need to show that it is always possible to do this, which is typically done by a construction that maps the members of $$A\,$$ and $$B,$$ into ordered pairs with different first members, such as 0 and 1, respectively. So, formally, we might have: Thanks to the role of isomorphism in category theory, we don't care whether we actually use this or any other systematic method for producing disjoint unions; this construction simply shows that there's always one to be found (proving that it works would be a drill in basic set theory).
 * $$A' = \{0\}\times A$$
 * $$B' = \{1\}\times B$$
 * $$\iota_1 = a \mapsto \langle 0, a \rangle : A\rightarrow A'\cup B'$$
 * $$\iota_2 = b \mapsto \langle 1, b \rangle : B\rightarrow A'\cup B'$$

And of course the motivation for the use of $$+,$$ to represent co-products is that for finite sets, if $$A\,$$ has $$m\,$$ members and $$B\,$$ $$n\,$$, then $$A+B$$ has $$m+n$$ members.

All this might be taken as pointing to one of the major basic failures of the 'new math' in the mid-to-late 20th century. I (AA) recall being told in perhaps third grade about sets and how they were the basis for everything, but they didn't seem to be able to do a decent job of explaining addition, due to the problem of what if the two sets being added overlapped? Third graders are, I believe, not generally capable of understanding disjoint unions, co-products, and the benefits of regarding isomorphic objects as identical (and I'd have my doubts about the teachers), so for people at that level, trying to present sets as the basis for everything is just a distraction from actually learning anything about numbers.

Exercise: In $$\mathcal Set$$, what is $$1+1\,$$? (naturally, it's called $$2\,$$, but what is it?)

Cartesian Categories and Deduction
An elementary but somewhat extended example of products (and coproducts) is logical conjunction (and) and disjunction (or). We develop this example in stages. It's a more relaxed presentation of material from Lambek & Scott (1986) Introduction to Categorical Logic Cambridge University Press.

[STILL IN NEED OF SOME CRACKFILLING, ETC]

Proofs as Morphisms
A basic idea of logic is that of logical consequence, whereby we have a conviction that, if we accept some proposition $$A\,$$, we should also accept some proposition $$B\,$$. For example, if we accept Bert is happy and Grover is sad, we should also accept Bert is happy. Logic gets going when we can begin to individuate these convictions into distinct patterns of argument, and give them names, so that solid arguments that will survive extensive criticism can be systematically identified and distinguished from delusions. The pattern of argument we just called for can be called 'case 1 of conjunction elimination', and basically says that if we accept a proposition of the form $$A\,$$ and $$B\,$$, we should also accept one of the form $$A\,$$. Aside from the details of what basic patterns of argument there are, two rather plausible logical principles are: This is getting close to constituting a category, where the propositions are the objects and the deductions the arrows, but to go all the way, we need to impose some equivalences, in particular those for the identities, and associativity. Let $$f:A\rightarrow B$$ mean that we have a deduction (proof) of $$B\,$$ from $$A\,$$. Then we require: which are just the basic equations of category theory.
 * given $$A\,$$, we can conclude $$A\,$$ (for every proposition $$A\,$$, there is an identity deduction $$1_{A}\,$$ of that proposition from that proposition)
 * if there is a deduction $$f\,$$ of $$B\,$$ from $$A\,$$, and $$g\,$$ of $$C\,$$ from $$B\,$$, then there is a deduction $$g\circ f$$ of $$C\,$$ from $$A\,$$ (transitivity of deduction)
 * $$1_B\circ f = f = f\circ 1_A$$
 * $$(h\circ g)\circ f = h\circ(g\circ f)$$

If we consider the deductions to be actual arrangements of marks on a medium such as paper or a chalkboard, this equivalences might hold 'automatically' (for example by pieces of deductions being combined without parentheses or other delimiters), or by conventions to the effect that certain combinations of symbols are deemed identical. Be this as it may, Lambek and Scott (1986) call a system with composition and identities a 'deductive system', which will be a category if the equations above are obeyed.

Conjunction as Products
Logic doesn't get very far without connectives, that is, means for combining propositions and perhaps ingredients of propositions to make more propositions. One of the more basic connectives is 'and', which, in terms of deductions, intuitively obeys these three rules (among others): We can formalized these intuitions as part of a deductive system by introducing a connective $$\wedge$$ for propositions, and assuming the following principles for constructing deductions: These deductions can exist independently of whether or not the identity and associative equations hold; we'll call a deductive system where they exist a 'conjunction system' (almost a conjunction calculus in the sense of Lambek and Scott, but this latter notion requires some more ingredients).
 * if we accept $$A\,$$ and $$B\,$$, we feel compelled to accept $$A\,$$.
 * if we accept $$A\,$$ and $$B\,$$, we feel compelled to accept $$B\,$$.
 * if we feel compelled to accept $$A\,$$ on the basis of $$C\,$$, and $$B\,$$ on the basis of $$C\,$$, then we feel compelled to accept $$A\,$$ and $$B\,$$ on the basis of $$C\,$$.
 * For propositions $$A, B\,$$, there exist the following two deductions:
 * $$\pi^1_{A,B}\colon A\wedge B\rightarrow A$$ (the first projection/case of conjunction elimination).
 * $$\pi^2_{A,B}\colon A\wedge B\rightarrow B$$ (the second projection/case of conjunction elimination).
 * If there are deductions $$f\colon C\rightarrow A$$ and $$g\colon C\rightarrow B$$, then there is a deduction $$\langle f,g\rangle \colon C\rightarrow A\wedge B$$.

So you can perhaps perceive the looming idea that if the deductive system is a category, conjunction and the projections might constitute a product. But for this to be the case, some equations will have to hold. These are:

For all $$f\colon C\rightarrow A$$, $$g\colon C\rightarrow B$$, $$h\colon C\rightarrow A\wedge B$$: In the first two equations I've included the subscripts to the projections, while in the last, I've left them out; you should be able to write them in on the basis of the environment.
 * $$\pi^1_{A,B}\circ\langle f,g\rangle=f$$ (ProjEq1).
 * $$\pi^2_{A,B}\circ\langle f,g\rangle=g$$ (ProjEq2).
 * $$\langle\pi_1\circ h, \pi_2\circ h\rangle = h$$ (PairEq).

These equations are sufficient for $$(A\wedge B, \pi^1_{A, B}, \pi^2_{A, B})$$ to constitute a product.

Proof: Suppose we have a conjunction system such that the above equations hold, and we have some $$D\,$$ such that there are deductions $$f\colon D\rightarrow A$$ and $$g\colon D\rightarrow B$$. We need to show that there is a unique deduction $$h\colon D\rightarrow A\wedge B$$ such that $$f=\pi^1\circ h$$ and $$g=\pi^2\circ h$$; we'll call these two equations the 'factoring requirements', the other requirement on $$h\,$$ being the uniqueness requirement.

Now by the rules for a conjunction system, we have $$\langle f,g\rangle\colon D\rightarrow A\wedge B$$, and from the equations ProjEq1 and ProjEq2, it follows that $$\langle{f,g}\rangle\,$$ satisfies the factoring requirements for $$h\,$$. Now suppose we also have $$k\colon D\rightarrow A\wedge B$$ that satisfies the factoring requirements. Then we have: which proves uniqueness. $$\Box$$
 * $$\langle{f, g}\rangle = \langle{\pi^1\circ k, \pi^2\circ k}\rangle = k $$

Exercise: Prove that 'conjunction is commutative', in the sense that in a conjunction system, there is a deduction of $$B\wedge A$$ from $$A\wedge B$$.

Exercises

 * 1) Explain following diagram and prove that $$h_f = h_g\ $$.[[image:Distributivity diagram.svg]]
 * 2) Prove that $$(A \cap B) \cup (A \cap C) \subseteq A \cap (B \cup C)$$.
 * 3) Prove that gcd(lcm(a,b), lcm(a,c)) is divisible by lcm(a, gcd(b,c)) for any positive integers a,b,c.
 * 4) Let $$A_1, A_2, A_3, \ldots\;$$ be objects in category $$\mathcal{C}$$ with all finite products. Prove by induction that $$\prod_{1 \le i \le n} A_i$$ is isomorphic to $$(((A_1 \times A_2) \times A_3) \ldots ) \times A_n$$ for all positive integers n.
 * [[image:Prod chain.svg|center]]
 * Hint: Set $$B_n := (((A_1 \times A_2) \times A_3) \ldots ) \times A_n$$ and $$C_n := \prod_{1 \le i \le n} A_i$$.
 * Prove that $$B_1 \cong C_1$$.
 * Assume you have proved $$B_n \cong C_n\;$$, and prove that $$B_{n+1} \cong C_{n+1}\;$$.

formulation of n-ary product
for $$n>0$$: An n-ary product of $$A_1,\ldots,A_n$$ is an object $$P\,$$ and arrows $$\pi_1,\ldots\pi_n$$ from $$P\,$$ to $$A_1,\ldots,A_n$$, respectively, such that for any object $$C\,$$ and arrows $$f_1,\ldots,f_n$$ from $$C\,$$ to $$A_1,\ldots,A_n$$, respectively, there is a unique arrow $$h=:C\rightarrow P$$ such that $$f_i=\pi_i\circ h$$, for $$i=1,\ldots n$$.

unary product diagram
[TO COME]

union problem
Let $$f\,$$and $$g\,$$ return different values for some member of $$A\cap B$$.

Related resources

 * Product (category theory)
 * Coproduct
 * Distributive category
 * Jocelyn Paine's interactive demonstrations about products and coproducts in the category of finite sets


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

title: Category:BCP/SEPA/11


text: Category:BCP/Sessile leaves

title: Category:BCP/Sessile leaves


text:
User talk:TheElvenLord

Welcome
Hello TheElvenLord, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! Erkan Yilmaz ( talk ?, wiki blog ) 13:35, 1 November 2007 (UTC)

Already working - great
Hi, about this: since I saw you made since then some edits, I guess you are already acclimated here a little. If you need anything, tell me please. See you around, Erkan Yilmaz ( talk ?, wiki blog ) 18:50, 2 November 2007 (UTC)

I've tidied up some or your layout a bit and done some copy editing (English and Cornish). Nyns eus ID genev omma na hwath, drog yw genev. Keep up the good work, once someone starts it others will come along and improve it (mostly). Sewen! Perderi 20:25, 3 November 2007 (UTC)

TEL, try to lay off the RanDoM caPiTal letters, and you don't need a space before commas etc. :-) Perderi 04:33, 5 November 2007 (UTC)

delete?
do you mean that Brythonic Celtic Langauges Division can be deleted because this has no typo Brythonic Celtic Languages Division ? Next time you can also move pages, instead recreating. Erkan Yilmaz ( talk ?, wiki blog ) 19:49, 7 November 2007 (UTC)
 * the one with incorrect title was deleted --Gbaor 11:44, 30 May 2008 (UTC)

page creation
I saw this. You should be able to create new pages now. Here is a place to start if you have never started a new page. If you need more help, please let me know. --JWSchmidt 18:10, 11 November 2007 (UTC)

ping
User_talk:Erkan_Yilmaz, Erkan Yilmaz ( talk ?, wiki blog ) 19:22, 14 November 2007 (UTC)
 * User talk:91.141.8.56: then people new to Wikiversity could immediately see the different spelling systems and do not make the same trouble making actions again. What do you say ? Erkan Yilmaz ( Wikiversity:Chat, wiki blog ) 17:52, 19 November 2007 (UTC)

Demad dit !
Hi,

Thanks for your proposal. I will be more than happy to help.

A galon.--Luzmael 09:09, 16 November 2007 (UTC)

more info
Please provide more information to this request for deletion: Kernewek Onan/Lesson:Grammar Rules of Cornish. You can (should) also use the Please delete my page function/page for in these cases. --Gbaor 11:49, 30 May 2008 (UTC) the same for this: Kernewek Dew/Lesson 3, was it moved somewhere? Please use the Please delete my page, while your request is more complex. --Gbaor 11:52, 30 May 2008 (UTC)

title: User talk:TheElvenLord


text:
Art History/Reflections/Kristen and Elizabeth

Kristen: is in bold

Elizabeth: is in regular

October 10th,2007

What a day we had today. I was feeling really frustrated because I was sure that Elizabeth and I had such a great symbol for our project and it just got blown out of the water because of stupid Art store not functioning on the computer. To look for things referring to immortality and mortality, well I guess you'd have to start using synonymous to both of the words. I chose the topic because there is such a defining line between the two topics and I want that to be evident in our museum. That isn't too much to ask is it?

October 11th,2007

I was happy I got to present my extra thing on Venus today! I thought it was a really good thing to bring up since I know lots of people had seen it before. If we can find something like that, can't we find more things on our theme? I personally am NOT abandoning it!

November 1st, 2007

Well, seeing as both Elizabeth and I both have seperate accounts on Artstor, I've picked out 5 from the groupings that I really liked and want to use and she's picked out her 5. Now we just need to split down to the finals and see which pictures have the most strengths and will be best for us to present. I'm excited, I know ours are really good and all we need now is background information. We could possibly start our project today...

November 1, 2007 (Elizabeth): I really feel like I had a breakthrough today during class. As Kristen and I decided to each choose 5 pictures to present, I felt the need to research further. One of my pieces that I am very excited about is called "Heart Scarab of Hatnofer." Covered in hyroglyphics, this interesting piece was placed over the heart of the mummy (and as the Egyptians believed, the heart of the spirit.) My research began by taking a closer look at the piece. This obviously didn't work because I can't read hyroglypgics. After that I looked in the introduction. It mentioned the inscription was from the "book of the dead." After a deep search, and by googling Hatnofer, I found the passage. Egyptian belief of guarding the heart is a belief that I find to be most interesting...and I completely agree. A spirit should always be with you. Today I learned that by researching with other sources, you can find out handy information needed to better understand the piece you are observing. This project is a great idea, but I am concerned that we still don't have enough pieces yet...hopefully that will change. Right now, I am happy with the progress that we have made and the realizations and conclusions (to better understand art and why it is created) that comes hand and hand.

November 7th, 2007

I thought it was really interesting when Ms. Davis was talking about Japan and how things go there. I would have never expected that kids are the ones who clean the schools! But they probably do it at home as well, clean up after themselves, so I doubt any of the Japanese kids are lazy like us Americans -_-. Me and Sona were talking afterwards about how it really would be obvious to spot an American tourist in Japan if you had the chance. Try taking a bunch of rowdy kids over there and see what happens XD.

November 8th, 2007

I'm kind of dissapointed with myself today, because I feel I could have done a better job with that powerpoint. The thing is, I don't know where our thread is connecting our pictures and we have such little time. I really liked the picture with Bassus and Napoleon awakening to Immortality, I thought those were original good pieces. It's just that I don't always want to do the Egyptians when it comes to Immortality and Mortality. I don't want to fall where the lines cross because that's just it, they don't cross. You either die or you live. I think the information on Matthew Brady (I think that's his name) was EXTREMELY helpful though. That's more of what I'd see as mortality, at people seeing dead bodies on a field and being horrified at the sheer loss of life. You can capture more emotion with a photograph, so I'm definately looking that up.

November 15, 2007: (Elizabeth) Wow. I havent reflected in a while. I was having a few doubts about stringing our project together. After meeting with Mr. Bishop & Mrs. Davis, I learned that we don't have to worry about having a continual "thread" running through our pieces. This was an INCREDIBLE relief. Also, I found it very helpful to have one on one conferences because all of the teachers are so incredibly helpful. We got some ideas like researching Matthew Brady, as well as some pieces like "The Third of May." We went on Artstor yesterday and had trouble finding the requested pieces....we probably spelled them wrong. Anna and Sona mentioned that they put their pieces in a powerpoint to see how they look in a collection. This was such a good idea. We discussed the arrangement of our pieces with Mrs. Davis & Mr. Bishop yesterday and I think that we are going to arrange them from mortality to immortality. For example we may start with the Matthew Brady piece (I believe its something like The Battle of Antietam) which is a photograph of DEAD soldiers. This picture is probably the most powerful in the bunch. Anyway, since this picture respresents death, we will begin with it to create something like a "story" with our pictures. (How someone goes from being a mortal, to being immortal, or the "journey to the afterlife.) Either way after discussing the subject matter together we agreed that this would connect all of our pieces and we would know which ones were strong, and which ones to weed out. We were thinking of ending with the piece the Arcadian Sheperds, because like we discussed in class, the "sheperds" we not watching over their dead friend, but infact they were envying their friends achievement of immortality. The last picture (if we can find one that is broad and not to blatant) would be something relating to heaven or an angel...I will ask Mrs. Davis, Mrs. Barker, or Mr. Bishop about any ideas. This picture would symbalize the achievement of immortality, and show the person looking down upon the earth. I just talked to Mrs. Barker about this picture and she believes that it would need to be a broad picture that can be explained. A picture that I am very excited about is one that Alison showed me yesterday. It is the Garden of Earthly Delights by Bosch. This painting would fit into our theme because it has three panels that show how mankind, originally immortal, sinned and lost that privelidge. It also shows the path to descruction. The book I got the picture in (the big art history book) describes the picture as "Illustrating the origin, indulgence, and punishment, of sin." It does just that. The last pannel of the tryptich, is a painting of hell, the most disturbing picture I have probably ever seen. The other two show the course to hell. The first painting, on the left, represents the Garden of Eden. It represents the world before sin. The second painting, in the center, represents the world before the flood (after the first sin, this is the fall of makind.) Something that I found very interesting about this piece is the motive and belief of the artist, Bosch. Bosch, an artist in the Italian Renaissance, opposed the Italian belief that was expressed by other artists of that time. The other Italian artists believed in representing the "beauty of life" in their paintings and works, while Bosch represented the "fraility and wickedness of mankind." I find it very interesting that he views so strongly against the common belief, and would like to do some more research to find out why. (Could this have been influenced in his upbringing?) Off to do work, but while I'm thinking about it, we have found the most pictures with the most substance today during class in the books. I will reflect about them when we have gathered more information.

November 19th,2007

So my computer was dead as a doorknob but somehow I was succesfully able to bring a list together. I'm still a little iffy on Otto Dix's work specifically because I cannot find the image that I scanned and I have yet to find the book so I assume I will be asking Ms. Barker for help. Our images and the depth centered around them is SO good though. I feel that I can connect to almost every piece and that is definitely reassuring. Besides that, Elizabeth and I are working really well together and I'm not as nervous now about our final product.

November 27th,2007

I'm dying. Yep, this project is officially killing me with its omnipresent force in my life. How the heck are we going to even attempt to finish this? We haven't even started our paragraphs yet! I'm just stressing out, bigtime. I don't understand how I can juggle this massive workload by Friday. When I even attempt to try to think of the overload of work that I'm under just in this class I start to break out in a sweat. *Okay stop venting, venting done*. But we started today, Elizabeth's worked on the thesis and I'm putting in the pieces and piling on stuff for our summaries. We'll see how this all turns out in the end -_-.

November 28, 2007 (Elizabeth): I am in dyer need to reflect! Kristen and I have been working very hard on our project, but we feel pushed for time. I have written a very good summary for one of my paintings: "The Garden of Earthly Delights". It is written in almost a journalistic style...anyway, back on track: This project has really made me feel like a more intelligent person. I have noticed that lately it seems to be weaving throughout my other subjects. This project was a great idea, but it is also INCREDIBLY stressful for me. You see, I am one of those people that takes forever to write something because I revise it while Im writing it (...as I press the delete key...) and it takes me a while to write summaries that I feel comfortable with. Another problem that I have been facing is internet issues. No computer's internet in my house has been working lately, and when I get home every night at 7:30 from swimming and get the rest of my homework done, leaving time to dedicate to Art History it is usually 10, 11, 12 o-clock that I have to drive to my dad's office and work on it...which is making it hard to get things finished without being completely exhausted the next day. Besides that, I feel like we are doing a great job, and we really understand the subject. We have a creative way of organizing our works as a story...that almost gives it a "twist" making the project even more exciting. I should reflect more today, but at 11:07 on a school night, at my dad's office, I should probably be working on my summaries.

December 9, 2007 (Elizabeth): I am incredibly stressed. Originally I thought that nine weeks was too long to do this project. After all of this time, I feel like we could have used a little more time because we feel time crunched. I think that if you repeat the project next year, you could have more "check up" days and due dates. For example: 2 summaries are due monday, 2 summaries are due thursday....etc. This would prevent some stress from building up, and I also feel that it would keep us on track. On a happier note, I really like the Chinese book binding project we are doing. It really emphasises the fact that we are doing all of this project on our own. I think that once we are finished I will really grow to cherish the book that we made ourselves as well as the overall project, no matter how stressful it may be now. Im sorry that this reflection isnt as deep as it should be. I am having problems coming up with that kind of thinking when it comes down to the end of our project. I have completed summaries for: the heart scarab of hatnofer, the garden of earthly delights, immortality, the equestrian tomb, the unidentified service man, the arcadian shepards, and the roman sarcophagus. And have "revamped" or REdone the equestrian tomb summary and right now I am REdoing the thesis.

December 14th,2007

Wow, I can't believe it. We're done, we're actually done. Man, this whole process has been absolutely incredible. I can look back on my book and the things I did and be absolutely amazed. I really want to thank Mr. Bishop, and Ms. Barker. They both did an absolutely amazing job helping me and keeping me motivated. Also I really want to thank Ms.Davis for doing such a wonderful job really leading the class. It seems forever that I came in early in the semester unsure of exactly what was going on ^_^. I really hope more people participate in the course and I will highly encourage them to do so!

<3333 <-- Those are hearts just so you know :) Kristen


text:
Art History/Reflections/Connor and Rawson

november 1: Rawson: Conner and I have found our 5 pictures for the powerpoint presentation. I believe that it could be difficult to create an entire powerpoint with citations and descriptions in just one period in the lab, but I guess we have to. We have a good number of images, but have not narrowed them down all the way to fifteen yet.

november 14: Rawson: Conner and I presented our power point and I am still kind of struggling to slog through the pictures and dig deeper for more meaning in our pictures. I guess I feel like since we were not really sure exactly what type of images we were looking for that none of our images on art store dig deep at all. I feel like we are going to have to start all over and find more images that are not just illustrations of power and authority. So I guess i better get started.

November 14: Connor: After Rawson and I's powerpoint didn't go so well, we started to go deeper on many of the works we chose. We have been using Art Through The Ages, where we used to only use art stor. That book has been incredibly helpful in our research and in finding background information for the works, and have helped us go past that "it looks powerful." However, now that we are going deeper, it seems that we have to nearly start from scratch, as most of our old works were just of big statues and leaders looking like gods. It may be a bit difficult to find 20 or so good works that can go beneath the first, more superficial level.

November 15: Rawson: Connor and I have been using books instead of Art Store and we have definitely progressed after our poor powerpoint presentation, but having to start over i fear that conner and i will not be able to find over fifteen by tomorrow. Overall, I believe that the books have helped us to look deeper into our theme of Power and Authority and not just at illustrations of it. As of Today we only have around eight images that I am content with so we still need to double that number which could be tough. We are attempting to go deeper into our theme and I believe that the pictures that we have aquired are a better representation of digging to a new level.

November 15: Connor: After starting completely from scratch, almost exclusively using books, Rawson and I have found plenty of images. We originally thought that we wouldn't be able to find any images, but we alrady have 10 summaries. We have mostly used Art Through the Ages, but recently I have been using a book on Mayan Royalty that has been tremendously helpful.

November 19: Connor: Rawson and I had to scan in all the photos from every book, which normally wouldn't be too difficult, but we have to track down all the books we used, which were reshelved, when we thought that they were still going to be in the seminar room. With all this, we doubt that we are going to be able to hit the deadline.

November 19: Rawson: I have had to scan in all the photos from all the books we used. This is annyoing because most of the images we used were in books. Because of the time it took to find all the books and to fix the scanner (it didn't work at first) I feel like we are behind in our formatting.

November 28th- Connor: After being behind, I feel like we are finally starting to catch up. Rawson and I are finishing up the summaries and already have a page or so of the thesis done. I feel like it is more realistic that we will hit our goal.

November 29th: Rawson: Initially, I was worried the we would not be able to finish by the deadline, but we got a lot done yesterday and things are looking optimistic for us. I think some of our summaries are a little bit shorter than the example, but hopefully that is not too much of a problem. I still have two more summaries to do, Connor is finishing up the introduction, and we still have to do the Bibliography in the next two periods, but I think that we will be alright.

December 12: after weeks of frantic working... I finally finished the book. I think that it would be a good idea if the next class were to get more time to work on the project... I don't really feel like we had enough time to really do a thorough job on every summary.


text:
User talk:Lestatoo

Welcome
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title: User talk:Lestatoo


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User talk:Lizzie3116

Welcome
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User talk:Lojian~enwikiversity

Welcome
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User talk:Titanium~enwikiversity

Welcome
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Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Titanium. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Titanium~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

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User talk:Shadoahvision

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User talk:Manuj~enwikiversity

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Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Manuj. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Manuj~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

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Renamed
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User talk:Shanata

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User talk:Venkatesamk

Welcome
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User talk:Farishta~enwikiversity

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Hello Farishta, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
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Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Farishta. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Farishta~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

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Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:36, 17 March 2015 (UTC)

Renamed
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User talk:D. H. Goldfish~enwikiversity

Welcome
Hello D. H. Goldfish, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! --Cybiko123 13:51, 1 November 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called D. H. Goldfish. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name D. H. Goldfish~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:30, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:05, 19 April 2015 (UTC)

title: User talk:D. H. Goldfish~enwikiversity


text:
User talk:Fatherdom~enwikiversity

Welcome
Hello Fatherdom, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
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 * Explore our learning projects;
 * Browse our portals, schools, and research activities;
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! --Cybiko123 13:51, 1 November 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Fatherdom. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Fatherdom~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:36, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:08, 19 April 2015 (UTC)

title: User talk:Fatherdom~enwikiversity


text:
User talk:Grajeshkannan

Welcome
Hello Grajeshkannan, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! --Cybiko123 13:51, 1 November 2007 (UTC)

title: User talk:Grajeshkannan


text:
User talk:Rajeshkannan~enwikiversity

Welcome
Hello Rajeshkannan, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
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 * Browse our portals, schools, and research activities;
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! --Cybiko123 13:51, 1 November 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Rajeshkannan. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Rajeshkannan~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:48, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:18, 19 April 2015 (UTC)

title: User talk:Rajeshkannan~enwikiversity


text:
User talk:RajeshK

Welcome
Hello RajeshK, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! --Cybiko123 13:52, 1 November 2007 (UTC)

title: User talk:RajeshK


text:
User talk:TarunShrivas!

Welcome
Hello TarunShrivas!, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! --Cybiko123 13:52, 1 November 2007 (UTC)

title: User talk:TarunShrivas!


text:
User talk:Kelovy

Welcome
Hello Kelovy, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! --Cybiko123 13:52, 1 November 2007 (UTC)

title: User talk:Kelovy


text:
User talk:Bala 1ay03ec011

Welcome
Hello Bala 1ay03ec011, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
 * Take a guided tour and learn to edit;
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! --Cybiko123 13:52, 1 November 2007 (UTC)

title: User talk:Bala 1ay03ec011


text:
User talk:PS430~enwikiversity

Welcome
Hello PS430, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
 * Take a guided tour and learn to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research activities;
 * Read and help develop our community policies；or
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! --Cybiko123 13:52, 1 November 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called PS430. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name PS430~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:40, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:17, 19 April 2015 (UTC)

title: User talk:PS430~enwikiversity


text:
User talk:Shells

Welcome
Hello Shells, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; for it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! --Cybiko123 13:52, 1 November 2007 (UTC)

title: User talk:Shells


text:
Bloom Clock/Keys/Southeastern Pennsylvania/November

November
All plants blooming in November (57)

 * 1) White flowers (27)
 * 2) Pink flowers (22)
 * 3) Red flowers (14)
 * 4) Orange flowers (8)
 * 5) Yellow flowers (18)
 * 6) Green flowers (1)
 * 7) Blue flowers (6)
 * 8) Purple flowers (22)
 * 9) Brown flowers (2)
 * 10) Black flowers (1)


text:
Art History/Reflections/Yeshwanth and Emma

Emma and I presented our powerpoint. I think all five of our works reinforced our theme, some more so than others. Ms. Davis gave us some suggestions for futher research on some of our works, which I found helpful. At this point, I think we'll keep the five for our museum and begin searching for the rest of our images, either in books or selected from our Artstor group.

After finding nearly 5 pages of images on ARTstor, today Yeshwanth and I narrowed down our selections to five images. I'm reall pleased with the selections we made and we both think that this will be an excellent project for both of us. We have made a wide selection which includes non-Western art, in addition to many different medias, including tapestry, sculpture, photography and paintings. We have begun working on our citations for the artwork and will begin working on our Powerpoint tomorrow.


text:
Bloom Clock/Keys/New Hampshire/November

November

 * 1) All plants recorded blooming in November in New Hampshire

title: Bloom Clock/Keys/New Hampshire/November


text:
User:Erkan Yilmaz/observations/2007 November

observations from Erkan Yilmaz for 2007 November

You are invited to add your comments (e.g. but my perception of this issue is different) - please sign anything you add (howto Signature).

2007 November 27 - collecting content from other wikimedia projects

 * There exists now a learning project at German WV: de:Projekt:Fundstücke aus anderen Wikimedia-Projekten
 * See also here at 2007 October 6 for the Wikimedia Garbage Detail

edit 2011-10-12:
 * deletions are a very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very controversial topic
 * see also

2007 November 17 - Greek Wikiversity - final approval ? & paper about WV

 * Greek Wikiversity final approval ?
 * Unfortunately not all comments are available.
 * edit 2007-11-20: seems not good or ?

 * paper about WV:
 * Today there was someone in the German chat who is going to write about Wikiversity in a seminary called "Seminar zu Design und Analyse von Web 2.0 Projekten" (there will also be a topic about Wikibooks). Let's see what will come out.
 * Was the one who will write about Wikibooks also there? Or was he/she in the wb channel? Do you know that? It seems to me that their tutor (Tischler) likes wiki and open source. :) -- 89.50.54.64 aka heuler06 17:14, 21 November 2007 (UTC)
 * Only de:Benutzer:Franz was at #wikiversity-de - perhaps he could give you details about the other person ? Erkan Yilmaz ( Wikiversity:Chat, wiki blog ) 17:53, 21 November 2007 (UTC)
 * This is interesting - will they be reporting back somehow, or will we have to nominate someone to keep tabs on the seminar? (Will that someone be you, Erkan? :-)) Cormaggio talk 09:59, 12 December 2007 (UTC)
 * Since the seminar will be in 1 month, I guess he will soon start. I have contacted him also now. I just saw a rough draft some time ago in the chat. Other questions of him can be found here (in German). We offered him help - regarding information - but of course he should do the work for himself :-) Erkan Yilmaz ( Wikiversity:Chat, wiki blog ) 17:03, 12 December 2007 (UTC)
 * The presentation and the paper (29 pages) are available now in German language, Erkan Yilmaz ( Wikiversity:Chat, wiki blog ) 17:37, 11 February 2008 (UTC)

edit 2011-10-21:
 * that guy never got active at de.WV anymore
 * as I see last edit on page de:Wikiversity:Presse was in 2009
 * I thought en.Wv would be better: In the media but the last edit there was even earlier :-(
 * outreach wikimedia has not more though: Bookshelf/Wikiversity#Wikiversity

2007 November 11 - what will happen if you start talking to yourself ?
Let's see what happens: incubator:User_talk:Erkan_Yilmaz
 * you are bored too much ;-) -- 83.133.5.85 aka heuler06 07:47, 15 November 2007 (UTC)
 * Hello heuler06, that was not the expected reaction I thought would happen, but this reaction is better than nothing. Erkan Yilmaz ( Wikiversity:Chat, wiki blog ) 11:25, 17 November 2007 (UTC)
 * We all talk to ourselves! A blog can be a running conversation with ourselves - which often hopes to include others in that conversation. As to "what will happen?", that can actually be quite a profound question, with profound answers. But as to whether we need to ask ourselves 'how we are', well, I think that may be pushing the idea of a conversation. ;-) Cormaggio talk 10:04, 12 December 2007 (UTC)

edit 2011-10-21:
 * yeah, sometimes I can be crazy, though others say that's being creative
 * I can recommend blogs, see also: User:Erkan_Yilmaz/PLE

2007 November 10 - Wikiversity flyer

 * de.WV flyer:
 * I hope we can finalize the German Wikiversity flyer proposal soon (hopefully still this year :-)). Let's awaken people again.
 * edit 2007-12-16: I think today we made one big step forward towards the realization of the German WV flyer. The salutation issue was decided (will be used the masculine form, though we have still different opinions, including me). For version 2 of the flyer we can optimize again.

edit 2011-10-21:
 * for such things meeting in real life is better
 * caused too much effort
 * what happened afterwards?
 * WMF Germany produced one without any community interaction :-(
 * see Colloquium/archives/August_2011

2007 November 5 - for some people I will suffer freely :-)

 *

2007 November 4 - no activity by users from universities, so what can we do about this?

 * de.WV recently had some edits regarding a course about Artificial Intelligence from an university. Plans were to use Wikiversity as a web platform besides the real life lecture. Unfortunately there are no edits anymore :-) So there were 2 people contacted from the team to get more feedback.
 * edit: Fast reply by the responsible persons. They will join again next year with another course. See you then.
 * Contacted also another responsible from another university who was active in April at de.WV

edit 2011-10-21:
 * the situation at de.WV in regards to universities is better now
 * though that also paralyzed the open nature of de.WV :-(

2007 November 2 - throw some chocolate into the German Wikiversity forum
...and let's see what will happen :-)

edit 2011-10-21:
 * I guess they though I'm crazy ;-)
 * as far as I remember was more positive than negative

2007 November 1 - how to revive the de:Wikiversity:Chat ?

 * how to revive the de.WV chat ?
 * Today I have included in my signature the link to the de:Wikiversity:Chat. I hope with this to increase peoples attention to it and coordinating things might get easier then. We are not so many at the moment at de.WV. I hope people can ask directly then for assistance and get others to participate in things they normally just scroll by in the Recent changes list. This is the first step, let's see if it will work.

 * edit 2008-03-17: changed my signature like this:
 * en.WV: --Erkan Yilmaz uses the Wikiversity:Chat (try)
 * de.WV: --Erkan Yilmaz benutzt den Wikiversity:Chat (versuch's mal)

edit 2011-10-21:
 * I have written at many places at de.WV + en.WV over the time + I guess this form of marketing is not so effective :-)


text:
User talk:Remi/Aug '07 – Oct '07

Concrete construction
Hi Remi. Thanks for your message. Probably you didn't get to see the content of the page in question, but it wasn't just Russian - it was auto-generated wiki-spam. If you want to create Concrete construction with welcome-and-expand, of course I have no objection, although perhaps you should reflect on how this page got created in the first place. One of the ways in which wiki-spam is generated is probably by spam-bots specially designed for MediaWiki. They are presumably fitted out with a long list of known MediaWiki installations. The wiki-spam-bot then identifies if IP users can create new pages or not. If not, it fakes a new user account. As a side point, wiki-spam-bots seems to create new never-active user accounts by the bucket-load just for fun anyway - possibly to make it more difficult to distinguish spam accounts from real ones. The spammer buys a flight of wiki-spam, probably specifying title and content. For each wiki, the bot fakes an IP, creates a one-use-only fake account if necessary, and creates a single new page. It probably iterates this over all the wikis on its current list, or a random subset.

What you should consider is that the page title was conceived by a spammer, not an educator, so it makes no particular sense for us to continue to use the title unless it is obviously educational.

-- McCormack 10:47, 24 August 2007 (UTC)

Nice catch
Have you voted yet, BTW? --SB_Johnny | PA! 11:12, 28 August 2007 (UTC)
 * I have the tabs open atm. Is this place like a democracy now? --Remi 11:14, 28 August 2007 (UTC)
 * Unfortunately it has to be in the CU case... meta makes the rules there :(. --SB_Johnny | PA! 11:15, 28 August 2007 (UTC)

:)
McCormack 14:03, 31 August 2007 (UTC)

Intro of Topic:Critical Pedagogy is a 99% copy of the WP article
Hello, could you have a look here please ? Thank you, Erkan Yilmaz ( evaluate me!, discussion ) 15:37, 3 September 2007 (UTC)
 * {{wps}) - Please make it better if you see that it can be improved. --Remi 09:31, 4 September 2007 (UTC)
 * Well I could only rephrase it, since I am not into that subject. But the context of any word could seem to a layman or even experts different. Rephrase the first sentence (see here please). Erkan Yilmaz ( evaluate me!, discussion ) 10:17, 4 September 2007 (UTC)

email
Could you activate the "email this user" feature for this account? Thanks. --JWSchmidt 22:31, 7 September 2007 (UTC)
 * I just did as per your request. Thanks for the reminder. --Remi 22:35, 7 September 2007 (UTC)

leaflet
Do you know what this is supposed to be? Wikiversity leaflet on Meta --JWSchmidt 04:01, 12 September 2007 (UTC)
 * Woops. I entered the link wrong. It should be here. Thanks. --Remi 04:50, 12 September 2007 (UTC)

Apptitude Test
You are invited to visit this project and explore/contribute to its parameters. Thank you, Shir-El too 15:46, 16 September 2007 (UTC)

WV as free hosting for political propaganda
Hi Remi. I see you've also been looking critically at the Palo Alto "learning project". The question has been running round my mind since this page was started: is it political propaganda, thinly disguised as a learning project, using WV as free hosting? What do you think? --McCormack 04:29, 23 September 2007 (UTC)
 * I think I know which page you are referring. I was simply moving a comment that was located in a page ending in "index.php" (a bug I think?) to an appropriate talk page. The opinions expressed by the author were not mine.
 * Personally, I have no opinion. I would need more information on the matter. I lean towards assuming good faith. --Remi 04:34, 23 September 2007 (UTC)
 * Yes - I realised afterwards you just moved an edit, and the edit meant something different anyway. The page is rather on the edge of my radar. --McCormack 04:43, 23 September 2007 (UTC)

What do you mean by "Sub pages"?
I am still not sure what people mean by sub pages? ~ Robert Elliott 16:06, 24 September 2007 (UTC)

Thanks!
...for the support at b:WB:RFA. – Mike.lifeguard | talk 17:19, 10 October 2007 (UTC)

Mentoring
Hi, Remi, I have a humble request for you. Would you be willing to mentor me as a probationary custodian? I'm willing to help out all I can - and really just need someone to get me going and a solid reference point should I encounter situations where I'm not too sure what to do (although this shouldn't happen to much after reading the guide and being well versed in Wikimedia projects). I run my own private installation of MediaWiki so the technical aspects shouldn't be a problem. It would be great if I could learn from someone obviously experienced and with the project's best interests at heart. Cheers, --Pumpmeup 06:15, 28 October 2007 (UTC)

title: User talk:Remi/Aug '07 – Oct '07


text:
Introduction to Functional Groups

Introduction
Functional groups are groups of atoms within a molecule that affect its function. Some basic ones are listed below.

Alkane
Every organic structure is based on alkanes. They contain only carbon and hydrogen, which are only connected through single bonds (C-H or C-C). These bonds are quite stable, non-polar, and therefore hydrophobic.

You can differ between linear, branched, and cyclic alkanes.

-list of linear alkanes with names, structure and semistructure will follow

Molecular Formula of Alkane is :CnH2n+2

Alkene
Alkenes are unsaturated hydrocarbons which have a general molecular formula of CnH2n. Hence, these are more reactive compared to their counterpart alkanes. Electrophilic addition reactions are a characteristic nature of alkenes. The first member is ethylene or ethene adds bromine to give 1,2-dibromoethane.

Alkyne
Triply bonded carbon atoms with the general structure CnH2n-2.

This is the SMILES for hexyne: H-C#C-C#C-C#C-H

Hydroxyl Group

The hydroxyl group is formed by attaching an oxygen and a hydrogen atom to a molecule. Both the bond from the molecule to the oxygen and the bond from the oxygen to the hydrogen are single bonds; thus hydroxyl groups are usually denoted R-O-H. (where O is oxygen, H is hydrogen, and R is the molecule the hydroxyl group is attached to)

An organic compound that contains a hydroxyl group is called an alcohol.

Reactions of alcohols is covered in Section 4.2

Carbonyl Group
The Carbonyl group is formed by attaching a oxygen to a carbon by a double bond, then attaching that same carbon atom to a molecule. A carbonyl group is denoted X-C=O

Carboxyl Group
The carboxyl group is formed by starting with a carbonyl group, then attaching a hydroxyl group to the same carbon atom in the carbonyl group. Finally, the carboxyl group is added to a molecule. Because carbon has four bonding sites, and three of them are taken up by the hydrogen and oxygen, the bond from the carbon to the molecule is a single bond. A carboxyl group is denoted X-COH=O

Aromatic
Aromatic compounds, also known as arenes or aromatics, are chemical compounds that contain conjugated planar ring systems with delocalised pi electron clouds instead of discrete alternating single and double bonds. Typical aromatic compounds are benzene and toluene. Considering benzene: Each carbon atom uses sp2 orbitals to form σ bonds to the other three atoms attached to it. This requires that all the atoms are in one plane. The carbon atoms have 1 electron in the p orbital which overlaps with its neighbours. This produces 6 new orbitals, three of which are bonding these are called π orbitals. Thus they form clouds of electrons above and below the ring in a torus shape and are known as the aromatic sextet. They can be represented as shown in figure 1. Figure 1: The aromatic sextet.

They should satisfy Hückel's law.

Heteroaromatic
Aromatic groups with more than one ring. An example of this is purine.

Haloalkane
Carbon-based motifs with a halogen, e.g. chloroethane.

Ether
R-O-R

Amine
R3-N, R2-NH, or R-NH2 are examples of amines.

Aldehyde
R-C(=O)-H

Ketone
R-C(=O)-R

Ester
R-C(=O)-O-R or H-C(=O)-O-R

Amide
R-C(=O)-N(-H)-R or R-C(-OH)=N-R Present in peptide bonds, which are bonds between amino acids in peptides and proteins.

Nitrile
R−C≡N This functional group is formed when nitrogen has a triple bond with carbon.

title: Introduction to Functional Groups