nLab cartesian monad

Cartesian monads

Cartesian monads


A cartesian monad is a monad on a locally cartesian category that preserves pullbacks and whose unit and multiplication are cartesian natural transformations.

Motivation through generalised multicategories

Ordinary categories can be defined as monads in the bicategory of spans of sets. Multicategories can be defined in a similar way. (A multicategory is like an ordinary category where each morphism has a list of objects as its domain, and a single object as its codomain; think vector spaces and multilinear maps).

To see how a multicategory CC can be defined as a monad in some appropriate bicategory, let C 0C_0 be the set of objects of CC, and notice that the domain of a morphism of CC–a finite list of objects–is an element of TC 0T C_0, where TT is the free monoid monad. In this way the data for CC can be conveniently organized in the diagram

C 1 d c TC 0 C 0.\begin{matrix} &&C_1&&\\ &\overset{d}\swarrow& &\overset{c}\searrow&\\ T C_0 &&&& C_0. \end{matrix}

Tom Leinster built on the idea of generalized multicategories, where the domain of a morphism can have a more general, higher dimensional shape than just a list. This is accomplished by considering a category \mathcal{E} other than Set\mathrm{Set}, and a monad TT on \mathcal{E}, and mimicking the above construction. So the data for a TT-multicategory is a diagram in \mathcal{E} like the one above.

To state the structure required on the data for a TT-multicategory, we want to define a bicategory in which the above span is an endomorphism. Then a TT-multicategory will be such a span together with structure making it a monad in that bicategory. The bicategory is (T)\mathcal{E}_{(T)}, the bicategory of TT-spans in \mathcal{E}. Its objects are the objects of \mathcal{E}, and its morphisms are spans

M d c TE E.\begin{matrix} &&M&&\\ &\overset{d}\swarrow& &\overset{c}\searrow&\\ T E &&&& E'. \end{matrix}

This won’t in general be a bicategory without a few extra assumptions. Identity spans are defined using the unit η:IdT\eta: \mathrm{Id}\to T. Composition of spans is defined using pullbacks and the multiplication μ:T 2T\mu: T^2\to T, so the category \mathcal{E} must at least have pullbacks–usually it will be finitely complete. The associativity and unit 22-cells are defined using the universal property of the pullbacks. However, these 22-cells won’t in general be invertible. In fact, it turns out that requiring the monad TT to be cartesian is exactly what is needed to ensure that the coherence 22-cells are isomorphisms, and hence that TT-spans do in fact form a bicategory. Maybe this should be the “fundamental theorem of cartesian monads”.

Extending Bénabou’s observation that a small category is a monad in the bicategory of spans of sets, Burroni defined TT-multicategories as monads in the bicategory (T)\mathcal{E}_{(T)} from above. When TT is the identity monad on Set\mathrm{Set}, TT-multicategories reduce to small categories, and when TT is the free monoid monad on Set\mathrm{Set}, TT-multicatories are exactly ordinary small multicategories.

As an indication of how this theory is useful as a language for higher categories, take TT to be the free strict ∞-category monad on the category of globular sets. Then TT-multicategories with exactly one object are called globular operads, and Leinster defines one such globular operad (the initial “globular operad with contraction”) for which the algebras are weak ∞-categories.



Let (T,μ,ν)(T, \mu, \nu) be a monad on a category CC. Specifically, T:CCT: C \to C is a functor, and μ:T 2T\mu: T^2 \to T and ν:Id CT\nu: \mathrm{Id}_C \to T are natural transformations, satisfying unital and associative axioms making TT a monoid in the (strict) monoidal category End(C)\mathrm{End}(C). This monad is cartesian if

  • the category CC has all pullbacks,
  • the functor TT preserves pullbacks,
  • the natural transformations μ\mu and ν\nu are cartesian. Recall that a natural transformation α:ST\alpha: S \to T between functors CDC \to D is cartesian if for each map f:ABf: A \to B in CC, the naturality square
    SA Sf SB α A α B TA Tf TB\array{ S A & \overset{S f}{\to} & S B \\ \alpha_A \downarrow & & \downarrow \alpha_B \\ T A & \underset{T f}{\to} & T B }

    is a pullback.


There is some slight inconsistency in the use of the word cartesian in category theory. Sometimes, a category is called cartesian if it has all finite limits; similarly, a functor is called cartesian if it preserves all finite limits. In most examples of cartesian monads, the category CC has a terminal object, and hence finite limits. However, the functor TT almost never preserves terminal objects. For example, the free monoid monad on Set\mathrm{Set} is cartesian, as can be checked directly, but T1T 1 \simeq \mathbb{N} is not a terminal object. In this sense, a cartesian monad is really locally cartesian.

Examples and Non-Examples

  • The free monoid monad () *:SetSet(-)^*: Set \to Set is cartesian.

  • The free category monad acting on quivers is cartesian.

  • The free strict ω\omega-category monad acting on globular sets, T:Set G opSet G opT: Set^{G^{op}} \to Set^{G^{op}}, is cartesian.

  • The free strict monoidal category? monad on Cat\mathrm{Cat} is cartesian.

  • The free symmetric strict monoidal category? monad on Cat\mathrm{Cat} – where all coherence cells are required to be identities except the symmetries ABBAA\otimes B \cong B \otimes A – is cartesian.


  • The free commutative monoid monad on Set\mathrm{Set} is NOT cartesian.

  • The free strict symmetric monoidal category? monad on Cat\mathrm{Cat} – where all the coherence cells are required to be identities including the symmetry isomorphisms ABBAA \otimes B \cong B \otimes A – is NOT cartesian. In fact this is exactly the free commutative monoid monad on Cat\mathrm{Cat}.


Relation to operads

To every non-symmetric operad, hence to every multicategory is associated a cartesian monad, such that the corresponding algebras over an operad coincide with the corresponding algebras over a monad.

  • Every p.r.a. monad is cartesian; these are sometimes called “strongly cartesian monads”.



Some past discussion about the term ‘cartesian’ has been moved to locally cartesian category.

Last revised on December 30, 2018 at 06:04:58. See the history of this page for a list of all contributions to it.