nLab cocategory




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

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

If c 0c 1c 0c_0 \to c_1 \leftarrow c_0 is a cocategory object in CC, then by homming out of c c_\bullet, one obtains a limit-preserving functor CCatC \to Cat. Under reasonable conditions, the adjoint functor theorem conversely implies that all limit-preserving functors CCatC \to Cat are obtained in this way.


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

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

  • In SetSet, every cocategory object is a coproduct of copies of the trivial cocategory object 1111 \to 1 \leftarrow 1 and the cocategory object 1211 \to 2 \leftarrow 1 where the two points are distinct. These represent the discrete category functor and the codiscrete category functors SetCatSet \to Cat, respectively.

Last revised on October 18, 2022 at 11:20:39. See the history of this page for a list of all contributions to it.