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What is the simplest way to check whether a given function of two arguments (Its arguments and the value are morphisms of some category.) is a direct product in categorical sense?

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What does it mean for a single morphism to be a direct product? – Zev Chonoles Jan 26 '12 at 16:25
@Zev Chonoles: From : "there exists a unique morphism f : Y \to X such that the following diagram commutes ... The unique morphism f is called the product of morphisms" – porton Jan 26 '12 at 16:31
I would call the corner $X_1\xleftarrow{\;\;\pi_1\;\;} X_1\times X_2\xrightarrow{\;\;\pi_2\;\;} X_2$ a direct product, not the morphism $f$, but I understand what you're asking now. You should indicate in your question the necessary conditions on the domains and codomains of the morphisms for it to make sense. – Zev Chonoles Jan 26 '12 at 16:36
@Zev Chonoles: In fact I have a function which takes two ARBITRARY (with arbitrary domains and codomains) morphisms of certain category. We may restrict this function to take only morphisms with identical domains. I suspect (after this restriction) it will be a direct product in categorical sense. – porton Jan 26 '12 at 16:50
up vote 1 down vote accepted

You seem to be asking whether it is possible to give an essentially algebraic axiomatisation of categorical products. The short answer is: yes, but you need some additional data.

Let $\mathcal{C}$ be a category. Suppose we have the following operations:

  • For every pair of objects $(A, B)$, another object $A \times B$ and two arrows $\pi_{A,B} : A \times B \to A$, $\pi'_{A,B} : A \times B \to B$.
  • For every triple of objects $(A, B, C)$ and pair of arrows $f : C \to A$, $g : C \to B$, an arrow $\langle f, g \rangle : C \to A \times B$, such that the following axioms hold:

    1. $\pi_{A,B} \circ \langle f, g \rangle = f$

    2. $\pi'_{A,B} \circ \langle f, g \rangle = g$

    3. For all $h : C \to A \times B$, $\langle \pi_{A,B} \circ h, \pi'_{A,B} \circ h \rangle = h$

Exercise. Verify that the triple $(A \times B, \pi_{A, B}, \pi'_{A, B})$ has the universal property of the product of $A$ and $B$.

Some other universal constructions in categories can also be made essentially algebraic: this is done in the first chapter of Lambek and Scott's Introduction to higher order categorical logic, for example.

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P.S. If $\mathcal{C}$ is a small category with all binary products, then it can be endowed with these operations. Otherwise one needs to assume a sufficiently large axiom of choice. – Zhen Lin Jan 26 '12 at 17:49
1. It seems that you've made a mistake. As far as I understand, it should be $\pi_{A,B} \circ \langle f, g \rangle = f$ not $\pi_{A,B} \circ f = f$. 2. Could you show how uniqueness follows? – porton Jul 22 '12 at 18:39
Uniqueness follows from the fact that the pairing is invertible. – Zhen Lin Jul 23 '12 at 2:15
OK, I read through your answer and confirmed it is equivalent to the standard definition. I wonder why your simplified description is missing in textbooks. – porton Jul 23 '12 at 8:29
I want to refer to your definition of direct product (not only finite but also infinite) in my research article. What should I write in my bibliography? – porton Jul 23 '12 at 8:30

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