Identifyng objects in a category

Question

Is there a general way of identifying objects in a category to produce a new category? Something like a quotient by a relation on objects. How would the morphisms behave?

Answer 1

There is a notion of a quotient category, but it deals with identifying sets of morphisms, and not the objects.

If $\mathcal{C}$ is a category, then we define a congruence relation $R$ on $\mathcal{C}$ to be a family of equivalence relations $R_{x,y}$ on $\hom_\mathcal{C}(x,y)$ , where $x,y$ are objects in $\mathcal{C}$, such that if $f_1,f_2:x\rightarrow y$ are two morphisms in $\hom_\mathcal{C}(x,y)$ such that $(f_1,f_2)\in R_{x,y}$ and $g_1,g_2:y\rightarrow z$ are two morphisms in $\hom_\mathcal{C}(y,z)$ such that $(g_1,g_2)\in R_{y,z}$, then $(g_1\circ f_1), (g_1\circ f_2), (g_2\circ f_1), (g_2\circ f_2)$ are all $R_{x,z}$.

Then we define the quotient category $\mathcal{C}/R$ as the category whose objects are those of $C$ and whose morphisms are the equivalence classes of morphisms in $C$. The composition of two equivalence classes is defined component-wise (i.e. $[f]\circ [g]=[f\circ g]$). The identity morphism in $\hom_{\mathcal{C}}(x,y)$ is equivalence class containing the identity morphism in $\hom_{\mathcal{C}/R}(x,y)$. It is easy to check by the requirements on the equivalence relations $R_{x,y}$ that these definitions satisfy the necessary laws to make $\mathcal{C}/R$ into a category.

Identifying objects would be difficult, because it wouldn't be immediately obvious how to make the relation work well with the morphisms, and there would issues with the composition (at the very least, with being meaningful; you could get rid of the non-identity morphisms altogether or make formal compositions).

Answer 2

It's not hard to compute a few examples of such quotients. Define $C/\sim$ to be the universal category with a functor from $C$ which is well defined on $~$-equivalence classes. This implies a colimit condition on the morphisms. For instance, if $C$ has two distinct objects, each with endomorphisms $\mathbb{Z}$ and with no morphisms between them, then the universal category identifying the two objects of $C$ is the category with one object and endomorphisms $F_2$, the free group on two letters. If instead $C$ had just one morphism between its two objects, and no endomorphisms but the identity, then identifying the two objects would produce $\mathbf{N}$. To get morphisms between distinct objects $[A],[B]$ of $C/\sim$, you'd take all morphisms $A'\sim B'$ with $A\sim A',B\sim B'$ from $C$ with free composition law, modulo the relations coming from $C$. Obviously this is sketchy, but I don't see any impediment to the construction-nor any hope that you'd be able to compute anything of interest in a significant number of cases.

Answer 3

Your question does not make too much sense, categorically speaking. That's because categories are basically collections of morphisms obeying certain composition rules (see for instance the definition given here, or "arrow only categories" in MacLane'CWM). Objects are an "afterthought" and correspond to identity morphisms. So , in this light, your question is: can we identify/merge identity morphisms? But different identities are just morphisms which have neither tip nor tail in common, so it is a bit "odd" trying to identify/merge them. What about other morphisms which have neither tip nor tail in common, for example?

What is interesting instead, and is done with congruences - as explained by Hayden in his answer - is to identify/merge morphisms which have both tip and tail in common (ie. they belong to the same Hom set).

Generally speaking, when trying to devise/understand a new construction in category theory, first try to see if it makes sense with morphisms and composition of morphisms, then it will automatically make sense with objects, since objects are nothing but a special kind of morphisms (identity morphisms).

Answer 4

It may be more convenient to only identify them up to isomorphism -- that is, one recipe for constructing a 'quotient' that 'identifies objects' is to add new morphisms to your category so as to make the two objects isomorphic.

Answer 5

Yes, there is indeed. There is a notion of a generalized congruences, which is an equivalence relation on objects and sequences of morphisms. The notion was first introduced in Generalized congruences - Epimorphisms in $\mathbf{Cat}$ by Marek A. Bednarczyk, Andrzej M. Borzyszkowski and Wieslaw Pawlowski (I found the notation in Emanuel Haucourts Categories of components and loop-free categories easier to read, though). These generalized congruences appear naturally when calculating certain colimits in $\mathbf{Cat}$, in particular coequalizers, which is described in the first reference.

If you're interested in quotients by group actions, you might wanna look at Group actions on posets by Eric Babson and Dmitry Kozlov.