# nLab cartesian monad

Cartesian monads

### Context

#### Higher algebra

higher algebra

universal algebra

# Cartesian monads

## 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 $\mathrm{Cat}$.

## Properties

### 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”.

## Discussion

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.