Idea: many categorical structures can be seen as categories whose morphisms have arities with a shape (and composition) specified by a monad \(T\).
Examples (being slightly loose in the meaning of \(T\)-category)
- categories are \(T\)-categories for \(T\) = identity monad
- multicategories ... \(T\) = free monoidal category monad
- virtual double categories ... \(T\) = free double category monad
- multisorted lawvere theories ... \(T\) = free category with finite products monad
Another intuition: \(T\)-categories are "loose \(T\)-algebras".
Some less obvious examples:
- topological spaces ... \(T\) = ultrafilter monad
- metric spaces ... \(T\) = identity monad on \([0,\infty ]\)-\(\mathsf {Mat}\)
- functors over a fixed \(\mathbb {C}\) ... \(T\) = free presheaf monad on \(\mathbf {Set}^{\textsf {ob}(\mathbb {C})}\)
Frameworks for generalized multicategories also provide a setting for general results on the passage between
coherent and universal structure, e.g. monoidal categories and multicategories, indexed categories and fibrations, etc.