suitable monad

Suitable monads


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

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

from TT-multicategories to TT-graphs has a left adjoint FF; this adjunction induces hence a monad T +=(UF,UϵF,η)T^+ = (UF, U\epsilon F,\eta) on C +:=(C,T)GraphC^+ := (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^+, T ++T^{++} and so on. This is used for example in some of the possible inductive definitions of opetopes.


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

  • CC is cartesian
  • CC has disjoint finite coproducts, which are also stable under pullback
  • CC has colimits of all nested sequences; they commute with pullback and have monic injections (coprojections)

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

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


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

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


Section 6.5 and appendix D in

Last revised on April 27, 2011 at 20:44:40. See the history of this page for a list of all contributions to it.