Inductive limit in category of (small) categories

Question

Recently I attended a talk where the speaker mentioned about inductive limit in the category of (small) categories. I have never seen anything like this so I was wondering how does one constructs inductive limit in category of (small) categories. More precisely, let $ C_1 \to C_2 \to \dotsb \to C_n \to \dotsb$ be an inductive system where each $C_i$ is (small) category and connecting maps are functors.

How does one constructs $\varinjlim C_n$

I am unable to see the possible objects of $\varinjlim C_n$. Any references or ideas?

Answer 1

The inductive limit $\varinjlim C_n$ can be computed via inductive limits of sets as follows.

Notation: For a small category $D$, let's write $\operatorname{ob}(D)$ and $\operatorname{mor}(D)$ for the sets of objects and morphisms of $D$ respectively.

We then have that $$\operatorname{ob}(\varinjlim C_n) = \varinjlim \operatorname{ob}(C_n)$$ and $$\operatorname{mor}(\varinjlim C_n) = \varinjlim \operatorname{mor}(C_n).$$ All of the other structure needed to describe the category (e.g. source and target functions) are similarly computed as inductive limits.

It's a good exercise to try to work out what the $\operatorname{Hom}$ sets of $\varinjlim C_n$ are from this description.

Warning: It is not true that general colimits of small categories can be computed in this way. This is only true for certain colimits.

Answer 2

Since you say Brian's answer isn't explicit enough for you (though it was perfectly clear–just not right for your level–) recall first that the inductive limit of a sequence of sets $S_1\to S_2\to \cdots,$ with maps $f_{ij}:S_i\to S_j,$ is the quotient of the disjoint union of the $S_i$ by the equivalence relation under which $x\in S_i,y\in S_j$ are equivalent if there is some $k\ge i,j$ such that $f_{ik}(x)=f_{jk}(y).$ (The fact that this relation is transitive is why this is an easier colimit to compute than many others: the indexing category is filtered.)

The content of Brian's answer is thus, first, that the objects of $\varinjlim C_i$ are the inductive limit of the objects of the $C_i.$ For the morphisms we have $(\varinjlim C_i)([x],[y])=\varinjlim_{i\ge j,k}(C_i(f_{ji}(x),f_{ki}(y))),$ where $x\in C_j$ and $y\in C_k.$ Here $[x]$ denotes the equivalence class of $x,$ $(\varinjlim C_i)([x],[y])$ means the homs in the category $\varinjlim C_i,$ while the second $\varinjlim$ is one of sets.

The most illustrative examples are when the functors $C_i\to C_{i+1}$ are faithful and injective on objects. Then $\varinjlim C_i$ is just the union. For instance, if $C_i$ is the poset $0<\ldots < i$ seen as a category and the functors $C_i\to C_{i+1}$ are the obvious inclusions, then $\varinjlim C_i$ is $\mathbb N.$ For an orthogonal example, if $C_i$ is the category with two objects $a,b,$ and $i$ parallel morphisms $a\to b$ (with no morphisms $b\to a$) then $\varinjlim C_i$ has objects $a,b$ and countably many parallel morphisms $a\to b.$ These examples show how this inductive limit is computed by repeating the computations one would do in the category of sets on both objects and morphisms.