Context

Higher algebra

higher algebra

universal algebra

Idea

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 $C$ 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

$\begin{matrix} &&C_1&&\\ &\overset{d}\swarrow& &\overset{c}\searrow&\\ T C_0 &&&& C_0. \end{matrix}$

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 $\mathrm{Set}$, 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

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

This won’t in general be a bicategory without a few extra assumptions. Identity spans are defined using the unit $\eta: \mathrm{Id}\to T$. 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.

Definition

Definition

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

• the category $C$ has all pullbacks,
• the functor $T$ preserves pullbacks,
• the natural transformations $\mu$ and $\nu$ are cartesian. Recall that a natural transformation $\alpha: S \to T$ between functors $C \to D$ is cartesian if for each map $f: A \to B$ in $C$, the naturality square
$\array{ S A & \overset{S f}{\to} & S B \\ \alpha_A \downarrow & & \downarrow \alpha_B \\ T A & \underset{T f}{\to} & T B }$

is a pullback.

Remark

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.

Examples and Non-Examples

• 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 $\mathrm{Cat}$.