nLab suitable monad

Idea

Given a cartesian monad $T$ on a locally cartesian category $C$ one can define a generalized graph and a generalized multicategory of type $T$ ($T$-multicategory). We want to impose a “suitability” condition on $C$ and $T$ which would ensure that one can talk about a free $T$-multicategory. In other words, the forgetful functor

$$U : (C,T)\text{-}Multicat \to (C,T)\text{-}Graph$$

from $T$-multicategories to $T$-graphs has a left adjoint $F$; this adjunction induces hence a monad $T^+ = (UF, U\epsilon F,\eta)$ on $C^+ := (C,T)Graph$. Moreover one wants to be able to build new cartesian monads iteratively, namely to ensure the same suitability conditions on the $T^+$, $T^{++}$ and so on. This is used for example in some of the possible inductive definitions of opetopes.

Definition

A category $C$ is suitable (in the sense of Tom Leinster) if

• $C$ is cartesian
• $C$ has finite disjoint coproducts, which are also stable under pullback
• $C$ has colimits of all nested sequences (i.e. $\omega$-chains of monomorphisms); they commute with pullback and have monic injections (coprojections)

A monad $T = (T,\mu,\eta)$ is suitable if

• $(T,\mu,\eta)$ is cartesian
• its underlying endofunctor $T$ preserves colimits of nested sequences

Properties

Any presheaf topos is suitable. Any finitary cartesian monad is suitable.

The forgetful functor from the category of $T$-multicategories to the category of $T$-graphs has a left adjoint and the adjunction is monadic; the category of $T$-graphs is then suitable and the induced monad is suitable.

Literature

Section 6.5 and appendix D in

• Tom Leinster, Higher operads, higher categories, London Math. Soc. Lec. Note Series 298, math.CT/0305049
• Tom Leinster, Generalized enrichment for categories and multicategories, math.CT/9901139

See also the appendix of:

• Brandon Shapiro, Enrichment of Algebraic Higher Categories, arXiv:2205.12235 (2022).