symmetric monoidal (∞,1)-category of spectra
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 can be defined as a monad in some appropriate bicategory, let be the set of objects of , and notice that the domain of a morphism of –a finite list of objects–is an element of , where is the free monoid monad. In this way the data for can be conveniently organized in the diagram
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 other than , and a monad on , and mimicking the above construction. So the data for a -multicategory is a diagram in like the one above.
To state the structure required on the data for a -multicategory, we want to define a bicategory in which the above span is an endomorphism. Then a -multicategory will be such a span together with structure making it a monad in that bicategory. The bicategory is , the bicategory of -spans in . Its objects are the objects of , and its morphisms are spans
This won’t in general be a bicategory without a few extra assumptions. Identity spans are defined using the unit . Composition of spans is defined using pullbacks and the multiplication , so the category must at least have pullbacks–usually it will be finitely complete. The associativity and unit -cells are defined using the universal property of the pullbacks. However, these -cells won’t in general be invertible. In fact, it turns out that requiring the monad to be cartesian is exactly what is needed to ensure that the coherence -cells are isomorphisms, and hence that -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 -multicategories as monads in the bicategory from above. When is the identity monad on , -multicategories reduce to small categories, and when is the free monoid monad on , -multicatories are exactly ordinary small multicategories.
As an indication of how this theory is useful as a language for higher categories, take to be the free strict ω-category monad on the category of globular sets. Then -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 ω-categories.
Let be a monad on a category . Specifically, is a functor, and and are natural transformations, satisfying unital and associative axioms making a monoid in the (strict) monoidal category . This monad is cartesian if
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 has a terminal object, and hence finite limits. However, the functor almost never preserves terminal objects. For example, the free monoid monad on is cartesian, as can be checked directly, but is not a terminal object. In this sense, a cartesian monad is really locally cartesian.
The free monoid monad is cartesian.
The free strict monoidal category? monad on is cartesian.
The free symmetric strict monoidal category? monad on – where all coherence cells are required to be identities except the symmetries – is cartesian.
The free commutative monoid? monad on is NOT cartesian.
The free strict symmetric monoidal category? monad on – where all the coherence cells are required to be identities including the symmetry isomorphisms – is NOT cartesian. In fact this is exactly the free commutative monoid monad on .
blog comment giving the Motivation above
Some past discussion about the term ‘cartesian’ has been moved to locally cartesian category.