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I went to a lecture on category theory, and I am trying to understand something: I want to know what it means for a category to be distributive. We were given a homework problem, but I don't even understand what it means: Let $C$ be a category such that for all $X,Y \in C$ there exists a product $X \times Y$ and a coproduct $X \oplus Y$ in $C$. Show that there is a canonical morphism

$\psi : (X \times Y) \oplus (X \times Z) \to X \times (Y \oplus Z)$. When $\psi$ is an isomorphism, $C$ is called a {distributive category}.

What does "Show there is a canonical morphism $\psi : (X \times Y) \oplus (X \times Z) \to X \times (Y \oplus Z)$ mean? What does "canonical morphism" mean? By the way, functors have not yet been introduced in the course. So far we have only learned the definition of category, product, and coproduct.

The homework was already due, so I did not turn this in. At this point, I have no opportunity to turn something in, and I am just trying to understand.

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Please do not ask the same question here and on MO simultaneously. –  Mariano Suárez-Alvarez Sep 23 '11 at 2:10
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Or at M.SE and MO simultaneously ... :) I've answered, because defining 'canonical' is interesting, but also voted to close, because the question is homework. –  David Roberts Sep 23 '11 at 2:26
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@David Roberts: why does it matter that the question is homework? It seems something like insisting that the asker have a soul. You can't really check, and it shouldn't matter. –  Anton Geraschenko Sep 23 '11 at 19:31
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closed as too localized by David Roberts, Theo Johnson-Freyd, Mariano Suárez-Alvarez, Gjergji Zaimi, Yemon Choi Sep 23 '11 at 4:08

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2 Answers

Canonical means that there exists something (in this instance a map) uniquely determined by the data given. There might be other maps, but this one exists without having to make choices. For the case at hand, the universal properties of products and coproducts (which should have been given to you) imply that there is a canonical map.

Note that 'canonical' is not generally a mathematically rigorous term, but works in the 'meta-language', in this case English.

This may not help you, Alison, but for others there is the nLab page canonical morphism which makes an attempt at formalising some aspects of the concept.

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It means induced by a universal property. In this case from either the product or coproduct.

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I do not know the term "universal property," nor do I know what you mean by "induced by". Could you give me a precise answer without added terminology? –  Alison Sep 23 '11 at 1:52
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Alison: maybe you should way until you have collected bit more knowledge to ask this question (thereby aboiding imposing strange limitations on the answerers... which are rather not the idea of MO, in fact) If you simply ignore the adjective canonical, then you will not miss much. –  Mariano Suárez-Alvarez Sep 23 '11 at 2:13
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