Limits and colimits

 Lecture topic

Definition: Given categories $A,B$ and functors $\mathcal F,\mathcal G:A\to B$, a natural transformation $\eta:\mathcal F\to \mathcal G$ is a collection of elements $\eta_X\in \Hom_B(\mathcal F(X),\mathcal G(X))$ for all $X\in \Obj(A)$ such that the diagram

commutes, whenever $f\in \Hom_A(X,Y)$.

Definition: For $X\in \Obj(A)$, define the constant category $\underline X$ to be the category with $\Obj(\underline X)=\{X\}$ and $\Hom_{\underline X}(X,X)=\{\id_X\}$. For any other category $B$, this may also be viewed as a natural transformation $\underline X:B\to A$ with $\underline X(Y)=X$ and $\underline X(f)=\id_X$ for any object $Y$ and any morphism $f$ of $B$.

Definition: Let $A$ be a small category and $\mathcal F:A\to B$ a functor. The colimit $\text{colim}(\mathcal F$) of $\mathcal F$ is an object $\text{colim}(\mathcal F)\in \Obj(B)$ and a natural transformation $\iota:\mathcal F\to \underline{\text{colim}(\mathcal F)}$ that is initial among all such natural transformations. We write $\iota_X:\mathcal F(X)\to \text{colim}(\mathcal F)$ and have $\iota(f)=\id_{\text{colim}(\mathcal F)}$ for any morphism $f$ of $A$.

In other words, whenever $Z\in \Obj(B)$ and $\eta:\mathcal F\to \underline{Z}$ is a natural transformation, there is a unique map $\zeta:\text{colim}(\mathcal F)\to Z$ such that the following diagram commutes:

Definition: Let $A$ be a small category and $\mathcal F:A\to B$ a functor. The limit $\lim(\mathcal F$) of $\mathcal F$ is an object $\lim(\mathcal F)\in \Obj(B)$ and a natural transformation $\pi:\underline{\lim(\mathcal F)}\to \mathcal F$ that is final among all such natural transformations. We write $\pi_X:\lim(\mathcal F) \to \mathcal F(X)$ and have $\pi(f)=\id_{\lim(\mathcal F)}$ for any morphism $f$ of $A$.

In other words, whenever $Z\in \Obj(B)$ and $\epsilon:\underline{Z}\to \mathcal F$ is a natural transformation, there is a unique map $\theta:Z\to \lim(\mathcal F)$ such that the following diagram commutes:

Examples of colimits are initial objects, coproducts, cokernels, pushouts, direct limits. Examples of limits are final objects, products, kernels, pullbacks, inverse limits.

 Remark: $\Hom$ commutes with limits and tensor commutes with colimits. That is:
\[
\Hom(A,\lim(B_i)) = \lim\left(\Hom(A,B_i)\right)
\hspace{1cm}
(\text{colim}(A_i))\otimes B = \text{colim}(A_i\otimes B)
\]
References: May (A Concise course in Algebraic Topology, Chapter 2.6), Aluffi (Algebra: Chapter 0, Chapter VIII.1)