Colimit

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 cocone from $D$ consists of an object $C$ of $\mathcal{C}$ together with morphisms

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

(using composition ).

A colimit of $D$ is a cocone $(\mathrm{colim}\,D,\kappa_j)$ such that for every other cocone $(N,\nu_j)$ from $D$, there exists a unique morphism $m:\mathrm{colim}\,D \to N$ with

One writes $\mathrm{colim}\,D$ (or $\varinjlim D$). If a colimit exists, it is unique up to unique isomorphism .

Relationship to other constructions

• The dual notion is the limit .
• Equivalently, $\mathrm{colim}_{\mathcal{C}} D$ is $\lim_{\mathcal{C}^{op}} D^{op}$ in the opposite category .

Examples

Example (Coproduct)

If $J$ is the discrete category on two objects and $D$ picks out objects $X$ and $Y$, then $\mathrm{colim}\,D$ is the coproduct $X\amalg Y$.

In $\mathbf{Set}$, this is the disjoint union of sets.

Example (Coequalizer)

If $J$ is the “parallel pair” shape $A \rightrightarrows B$, then $\mathrm{colim}\,D$ is the coequalizer of the two morphisms.

In $\mathbf{Set}$, this is a quotient $B/{\sim}$ identifying $f(a)\sim g(a)$.

Example (Pushout)

If $J$ is the span shape $X \leftarrow Z \rightarrow Y$, then $\mathrm{colim}\,D$ is the pushout $X\amalg_Z Y$.

In $\mathbf{Top}$, this is the usual gluing construction obtained as a quotient of $X\amalg Y$ by identifying $f(z)$ with $g(z)$.