# Merging / combining categories

Given a category $\mathcal X$ (that will be used as an underlying category) and collection of categories $\mathcal C(i)$ for $i \in I$ with a faithful functor from each $\mathcal C(i)$ to $\mathcal X$, a category $\mathcal C$ is called a fibered product of the diagram described if there is a faithful functor from it to every $\mathcal C(i)$ in a way that makes the diagram commute and it is a terminal object in the quasi-category of such sources / cones.

${\sf CAT}(\mathcal X)$ denotes the slice category of (the subcategory of ${\sf CAT}$ which contains only faithful functors) over $\mathcal X$.

1. I guess that a product in ${\sf CAT}(\mathcal X)$ is a fibered product in ${\sf CAT}$, right?
2. If so (*1), is ${\sf CAT}(\mathcal X)$ closed under products? / Is ${\sf CAT}$ closed under fibered product?
3. In the known cases of merged / combined categories such as: ${\sf TopGrp}$, ${\sf TopVec}$, (and even ${\sf Rng}$, ${\sf Grp}$ themselves are actually non-trivial combinations of categories), is the fibered product equal / isomorphic / equivalent / higher order something to the merge of its components?
• Please, consider using math symbols. This looks hard to read. – Fernando Muro Jun 11 '14 at 11:45
• An object in the pullback (strict or weak) in CAT of Top->Set<-Grp ist a set with a topology and a group structure. Nothing ensures that the structure maps of the group are continuous. Same with the required compatibilities in your other examples, e.g. if you try to present Rng as the pullback of Grp->Set<-Monoid you won't get a distributive law... – Peter Arndt Jun 11 '14 at 17:17
• Just FYI, although it probably won't change anything, these pullbacks involved with my question contain only faithful functors. – user52856 Jun 11 '14 at 21:36
• @PeterArndt , sorry for doing this in a double post, I didn't know that pressing an enter is so dangerous here, I pressed on the enter prior to asking what is the construction I should be interested about in this case? – user52856 Jun 11 '14 at 21:38
• @user52856 The combination of several structures is category theoretically best handled by looking at internal versions of the structures in question. Topological groups are group objects in Top. Group objects can be seen as product preserving functors from a small category, the Lawvere theory of groups, or from an even smaller product sketch. Combination of first order structures can be nicely described in the language of sketches: There is a tensor product of sketches such that a model of $A \otimes B$ is the same as a model of $A$ in the category of models of $B$... – Peter Arndt Jun 13 '14 at 0:33