nLab cocategory

Idea

A (small) cocategory is a category internal to $Set^{op}$, the opposite category of Set.

More generally, if $C$ is finitely cocomplete, there is a notion of cocategory internal to $C$, namely a comonad in the bicategory of cospans in $C$.

If $c_0 \to c_1 \leftarrow c_0$ is a cocategory object in $C$, then by homming out of $c_\bullet$, one obtains a limit-preserving functor $C \to Cat$. Under reasonable conditions, the adjoint functor theorem conversely implies that all limit-preserving functors $C \to Cat$ are obtained in this way.

Examples

• Many interval objects are cocategory objects. For example, the arrow category is a cocategory object in $Cat$.

• Any coalgebra object is a cocategory object. This includes corings, Hopf algebroids, cogroupoids?, etc.

• In $Set$, every cocategory object is a coproduct of copies of the trivial cocategory object $1 \to 1 \leftarrow 1$ and the cocategory object $1 \to 2 \leftarrow 1$ where the two points are distinct. These represent the discrete category functor and the codiscrete category functors $Set \to Cat$, respectively.

Related entries

• Trimble n-category