suitable monad

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

$U : (C,T)Multicat \to (C,T)Graph$

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.

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

- $C$ is cartesian
- $C$ has disjoint finite coproducts, which are also stable under pullback
- $C$ has colimits of all nested sequences; 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

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.

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 enrichement for categories and multicategories*, math.CT/9901139

Revised on April 27, 2011 20:44:40
by Toby Bartels
(64.89.61.127)