Limit - Knowlpedia

Let $\mathcal{C}$ be a category and let $J$ be an indexing category. A diagram of shape $J$ in $\mathcal{C}$ is a functor

D:J\to \mathcal{C}.

A cone to $D$ consists of an object $L$ of $\mathcal{C}$ together with morphisms

\lambda_j: L \to D(j)\quad (j\in \mathrm{Ob}(J))

such that for every morphism $\alpha:j\to k$ in $J$ ,

D(\alpha)\circ \lambda_j = \lambda_k

A limit of $D$ is a cone $(L,\lambda_j)$ such that for every other cone $(N,\nu_j)$ to $D$ , there exists a unique morphism $m:N\to L$ with

\lambda_j\circ m = \nu_j\quad\text{for all }j.

One writes $L = \varprojlim D$ (or $\lim D$ ). If a limit exists, it is unique up to unique isomorphism.

Relationship to other constructions

Examples

If $J$ is the discrete category on two objects and $D$ picks out objects $X$ and $Y$ , then $\lim D$ is the categorical product $X\times Y$ .

In $\mathbf{Set}$ , this recovers the usual cartesian product of sets.

If $J$ is the “parallel pair” shape $A \rightrightarrows B$ , then $\lim D$ is the equalizer of the two morphisms.

In $\mathbf{Set}$ , for functions $f,g:A\to B$ , the equalizer is the subset $\{a\in A\mid f(a)=g(a)\}$ .

If $J$ is the cospan shape $X\to Z \leftarrow Y$ , then $\lim D$ is the pullback $X\times_Z Y$ .

In $\mathbf{Set}$ , this is the fiber product $\{(x,y)\in X\times Y \mid f(x)=g(y)\}$ .