Examples of colimits in a category of categories

Question

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.

Answer 1

(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 $$\mathsf{Mod}(k) \to \mathsf{Mod}(k[t]/(t^2)) \to \mathsf{Mod}(k[t]/(t^3)) \to \dotsc$$