symmetric monoidal (∞,1)-category of spectra
A cartesian monad is a monad on a locally cartesian category that gets along well with pullbacks.
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 $C$ can be defined as a monad in some appropriate bicategory, let $C_0$ be the set of objects of $C$, and notice that the domain of a morphism of $C$–a finite list of objects–is an element of $T C_0$, where $T$ is the free monoid monad. In this way the data for $C$ 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 $\mathcal{E}$ other than $\mathrm{Set}$, and a monad $T$ on $\mathcal{E}$, and mimicking the above construction. So the data for a $T$-multicategory is a diagram in $\mathcal{E}$ like the one above.
To state the structure required on the data for a $T$-multicategory, we want to define a bicategory in which the above span is an endomorphism. Then a $T$-multicategory will be such a span together with structure making it a monad in that bicategory. The bicategory is $\mathcal{E}_{(T)}$, the bicategory of $T$-spans in $\mathcal{E}$. Its objects are the objects of $\mathcal{E}$, 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 $\eta: \mathrm{Id}\to T$. Composition of spans is defined using pullbacks and the multiplication $\mu: T^2\to T$, so the category $\mathcal{E}$ must at least have pullbacks–usually it will be finitely complete. The associativity and unit $2$-cells are defined using the universal property of the pullbacks. However, these $2$-cells won’t in general be invertible. In fact, it turns out that requiring the monad $T$ to be cartesian is exactly what is needed to ensure that the coherence $2$-cells are isomorphisms, and hence that $T$-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 $T$-multicategories as monads in the bicategory $\mathcal{E}_{(T)}$ from above. When $T$ is the identity monad on $\mathrm{Set}$, $T$-multicategories reduce to small categories, and when $T$ is the free monoid monad on $\mathrm{Set}$, $T$-multicatories are exactly ordinary small multicategories.
As an indication of how this theory is useful as a language for higher categories, take $T$ to be the free strict ∞-category monad on the category of globular sets. Then $T$-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 $(T, \mu, \nu)$ be a monad on a category $C$. Specifically, $T: C \to C$ is a functor, and $\mu: T^2 \to T$ and $\nu: \mathrm{Id}_C \to T$ are natural transformations, satisfying unital and associative axioms making $T$ a monoid in the (strict) monoidal category $\mathrm{End}(C)$. This monad is cartesian if
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 $C$ has a terminal object, and hence finite limits. However, the functor $T$ almost never preserves terminal objects. For example, the free monoid monad on $\mathrm{Set}$ is cartesian, as can be checked directly, but $T 1 \simeq \mathbb{N}$ is not a terminal object. In this sense, a cartesian monad is really locally cartesian.
The free monoid monad $(-)^*: Set \to Set$ is cartesian.
The free category monad acting on quivers is cartesian.
The free strict $\omega$-category monad acting on globular sets, $T: Set^{G^{op}} \to Set^{G^{op}}$, is cartesian.
The free strict monoidal category? monad on $\mathrm{Cat}$ is cartesian.
The free symmetric strict monoidal category? monad on $\mathrm{Cat}$ – where all coherence cells are required to be identities except the symmetries $A\otimes B \cong B \otimes A$ – is cartesian.
BUT:
The free commutative monoid? monad on $\mathrm{Set}$ is NOT cartesian.
The free strict symmetric monoidal category? monad on $\mathrm{Cat}$ – where all the coherence cells are required to be identities including the symmetry isomorphisms $A \otimes B \cong B \otimes A$ – is NOT cartesian. In fact this is exactly the free commutative monoid monad on $\mathrm{Cat}$.
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.
Tom Leinster, Higher Operads, Higher Categories (arXiv:math.CT/0305049), section 4.1
Albert Burroni, $T$-catégories (catégories dans un triple), Cahiers de Topologie et Géométrie Différentielle Catégoriques, 12 no. 3 (1971), p. 215-321 (numdam)
blog comment giving the Motivation above
Some past discussion about the term ‘cartesian’ has been moved to locally cartesian category.