nLab
cocategory

Contents

Idea

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.

Examples