# Examples of colimits in a category of categories

Can someone give an example of a colimit in a category of categories? In particular, it would be nice to see a well known category as a colimit of other well known categories. Describe the diagram of the cone as well.

I have another, similar question that is a bit more specific, and regards accessible categories.

• Coproducts are disjoint unions. Feb 25, 2015 at 19:34
• It's also true in the category of categories. Feb 25, 2015 at 20:02
• Groups is a category of categories, if you pretend that each group is a category with one object in the usual way, so you can just consider colimits there. Mar 2, 2015 at 7:38
• Hi Ben, you have mentioned a motivating example. Before I was thinking about colimits in a category of categories, I was thinking about groups presented with generators and relations. In the category you talk about, we see that the finitely presentable categories form diagrams. These diagrams have colimits which are other groups that are not finitely presentable. The arrow between groups goes from one group to another group with the same axioms plus one or more extra axioms. Thus, the finitely presentable groups are approximations to groups that have no finite presentation. Mar 2, 2015 at 23:32
• Your question reminds me this paper. They give some examples of coequalizers in $\mathbf{Cat}$. Mar 7, 2015 at 7:34

(1) The coproduct of a family of categories $$\mathcal{C}_i$$ is given by $$\mathrm{Ob}(\coprod_i \mathcal{C}_i) = \coprod_i \mathrm{Ob}(\mathcal{C}_i) = \bigcup_i \mathrm{Ob}(\mathcal{C}_i) \times \{i\}$$ $$\mathrm{Hom}_{\coprod_i \mathcal{C}_i}((X,i),(Y,j)) = \left\{\begin{array}{cl} \mathrm{Hom}_{\mathcal{C}_i}(X,Y) & i = j \\ \emptyset & i \neq j \end{array}\right.$$ and the unique composition rule such that the inclusions $$\mathcal{C}_i \to \coprod_i \mathcal{C}_i$$ become functors.

An explicit example is the category of fields, which decomposes as $$\mathsf{Fld} \cong \coprod_{p \in \mathbb{P} \cup \{0\}} \mathsf{Fld}_p,$$ where $$\mathsf{Fld}_p$$ denotes the category of fields of characteristic $$p$$.

(2) If $$\mathcal{C}_1 \xrightarrow{F_1} \mathcal{C}_2 \xrightarrow{F_2} \dotsc$$ is a sequence of functors, their colimit $$\varinjlim_i \mathcal{C}_i$$ may be described as follows: Objects have the form $$(X,i)$$ for some $$i \in I$$ and $$X \in \mathrm{Ob}(\mathcal{C}_i)$$ (!see the comment section for a correction!). The set of morphisms from $$(X,i)$$ to $$(Y,j)$$ is given by $$\mathrm{Hom}_{\varinjlim_i \mathcal{C}_i}((X,i),(Y,j)) = \varinjlim_{k \geq i,j} \mathrm{Hom}_{\mathcal{C}_k}(F_{i,k}(X),F_{j,k}(Y)),$$ where $$F_{i,j} : \mathcal{C}_i \to \mathcal{C}_k$$ is the composition $$F_{k-1} \circ \dotsc \circ F_i$$.

Here is an example: If $$k$$ is a field of characteristic zero, then one can show that the category of finite-dimensional algebraic representations of the additive group $$\mathbb{G}_a$$ over $$k$$ is isomorphic to the category of pairs $$(V,\phi)$$, where $$V$$ is a finite-dimensional vector space over $$k$$ and $$\phi$$ is a nilpotent endomorphism of $$V$$. This category is isomorphic to the colimit of the scalar restriction functors $$\mathsf{Mod}(k) \to \mathsf{Mod}(k[t]/(t^2)) \to \mathsf{Mod}(k[t]/(t^3)) \to \dotsc$$

• I think your second example is not quite correct -- it's probably equivalent to a colimit of categories, but not isomorphic. The class of objects you describe is just a disjoint union of objects at each level, but it should be the actual colimit of objects. (This can be seen e.g. by observing that the functor $\operatorname{ob} : Cat \to Set$ has a right adjoint given by mapping a set to a category with the set as objects and a unique morphism between any two object.) Mar 6, 2019 at 12:51
• Thanks! I made a short edit. Dec 22, 2019 at 14:40
• Please, what is a good reference for the sequential (or directed) colimit of categories?
– Tomo
Sep 16, 2021 at 1:43