Limit

Definition

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

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

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

(using composition ).

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

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

Relationship to other constructions

• The dual notion is the colimit .
• Many familiar constructions are special limits (see examples below).

Examples

Example (Categorical product)

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.

Example (Equalizer)

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 $\{a\in A\mid f(a)=g(a)\}$.

Example (Pullback)

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 $\{(x,y)\in X\times Y \mid f(x)=g(y)\}$.