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It is well known that in the category of all topological spaces, quotient maps aren't preserved by products (this follows from the simpler fact that $X\times (-):Top\to Top$ doesn't preserve quotients). The usual solution, if one is needed, is to change to the category of k-spaces and k-continuous maps. There are other categories where products and quotients 'get along' (e.g. $Set$, $Ab$).

Question 1: What is a large class of categories where quotient maps are preserved by products? Topoi? (Semi)abelian categories? Categories of algebras for a monad on a given category with this property?

Now one may be only interested in a certain class of quotient maps (like surjective submersions in $Diff$, the category of finite dimensional smooth manifolds). Say, regular epimorphisms, or maps admitting local sections (assuming we're in a site), or perhaps something like surjective topological submersions, where there are sections through every point in the domain. So in this case it is not a matter of putting restrictions on $X$ such that $X\times(-)$ preserves quotients, or changing the category, but narrowing the scope of the quotient maps one wants to preserve.

Question 2: Is there are large class of quotient maps (in $Top$, or in a general category - with finite products and enough colimits) that are preserved by products?

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up vote 3 down vote accepted

Note: I have edited this answer further because I was being silly before (unnecessarily restrictive).

I take it you mean categories $E$ for which $- \times -: E \times E \to E$ preserves quotients. The word 'quotient' may be slightly ambiguous because sometimes people use it to mean'coequalizer', and sometimes just 'epi' (as in 'quotient object'), but I take it you mean 'coequalizer'.

A reasonably large class would be regular categories, which includes categories of algebras of monads on $Set$ and semi-abelian categories and toposes. Here quotients = regular epis are stable under pullback and in particular are closed under taking products on either side. Furthermore, in a regular category, every quotient is a reflexive coequalizer, meaning a coequalizer of a pair $f, g: X \to Y$ for which there exists $h: Y \to X$ with $f \circ h = g \circ h = 1_Y$. In particular, the two projections $\pi_1, \pi_2: E \to Y$ of an equivalence relation $E$ on $X$, for example the kernel pair of a quotient, form a reflexive pair by the reflexivity property. So in a regular category, where quotients = coequalizers are necessarily coequalizers of their kernel pairs, quotients are quotients of reflexive pairs.

The reason reflexivity is relevant is a $3 \times 3$ lemma which says that in a (edit: commutative-in-parallel) diagram of $3 \times 3$ objects in which all rows and all columns are coequalizer diagrams of reflexive pairs, the diagonal is a coequalizer diagram. See the first page of Johnstone's Topos Theory. Then apply this lemma to the evident diagram whose rows are of the form

$$X_i \times X_{1}' \stackrel{\to}{\to} X_i \times X_{2}' \to X_i \times X_{3}'$$

and whose columns are of the form

$$X_1 \times X_{j}' \stackrel{\to}{\to} X_2 \times X_{j}' \to X_3 \times X_{j}'$$

In the category $Top$, it would therefore be natural to consider quotients by equivalence relations (or even just reflexive relations) which are preserved by taking products on each side. It's that latter condition which needs to be characterized (or at least discussed further), and I may come back to that later after I get the kids off to school. :-)

Edit: For a discussion of topological quotients which are stable under taking a product on either side, see the paper by Day and Kelly, On topological quotient maps preserved by pullback or product, Math. Proc. Cam. Phil. Soc. 67 (1970), 553-558. Or google Day-Kelly maps to find out more.

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Perhaps you mean to say that the functor $-\times -$ should preserve quotients/coequalizers in each variable separately? That's a different condition from its preserving coequalizers as a single functor on $E\times E$. –  Mike Shulman Oct 20 '10 at 17:31
Wait, I thought David was asking for the joint preservation of quotients. And I was trying to establish that: first, in a regular category, $X \times -$ does preserve quotient maps = regular epis (since the pullback of a regular epi $Y \to Z$ along the projection $X \times Z \to Z$ is a regular epi), and second, since regular epis are reflexive coequalizers, the 3 x 3 lemma in Johnstone's book guarantees that the fork down the diagonal is also a coequalizer; the corollary is the joint preservation. Have I made a stupid mistake? –  Todd Trimble Oct 20 '10 at 19:17
I was asking for preservation separately, but only under the illusion that this implied jointly. Really I'm interested in when the 'orbit space' of a product of internal groupoids is the product of the orbit spaces of the groupoids. Since a groupoid gives us a reflexive coequaliser, if reflexive coequalisers are preserved under products in a wider setting than coequalisers, then that would be great. –  David Roberts Oct 21 '10 at 0:42
@David: well, then your wish is granted! If you have separate preservation of reflective coequalizers, then you have joint preservation by this result. (I had a sneaking suspicion that your intended applications were topological or smooth, so that the regularity hypothesis wouldn't be all that apt for your purposes, but that hypothesis did address some of the other queries I believe.) –  Todd Trimble Oct 21 '10 at 0:53
Ah, but I don't know about separate preservation of reflexive coequalisers! I just found the result you mentioned (on the nLab of all places!) but that's only half of what I need. Is that 3x3 lemma applicable? To quote from Lawvere's paper CATEGORIES OF SPACES MAY NOT BE GENERALIZED SPACES AS EXEMPLIFIED BY DIRECTED GRAPHS, "..as is well-known, reflexive coequalizers preserve products, whereas irreflexive coequalizers do not." is this just in the specific context of topos theory or more general? Is it even applicable? –  David Roberts Oct 21 '10 at 2:10
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