# nLab semicartesian monoidal category

Semicartesian monoidal categories

### Context

#### Monoidal categories

monoidal categories

category theory

# Semicartesian monoidal categories

## Definition

A monoidal category is semicartesian if the unit for the tensor product is a terminal object, a weakening of the concept of cartesian monoidal category.

Many semicartesian monoidal categories are also symmetric, and sometimes that is included in the definition.

## Examples

Some examples of semicartesian monoidal categories that are not cartesian include the following.

• The category of Poisson manifolds with the usual product of Poisson manifolds as its tensor product.

• The opposite of the category of associative algebras over a given base field $k$ with its usual tensor product $A \otimes B$.

• The category of measurable locales, equivalently, the opposite category of commutative von Neumann algebras, equipped with the spatial tensor product, which corresponds to the measure-theoretic product of measurable spaces (but not the categorical product).

• The category Cat with the non-cartesian funny tensor product.

• The category of strict 2-categories with the Gray tensor product, and the category of strict omega-categories with the Crans-Gray tensor product.

• The category of affine spaces made into a closed monoidal category where the internal hom has $hom(x,y)$ being the set of affine linear maps from $x$ to $y$, made into an affine space via pointwise operations.

• The category of convex spaces, also known as ‘barycentric algebras’, made into a closed monoidal category where the internal hom has $hom(x,y)$ being the set of convex linear maps from $x$ to $y$, made into an barycentric algebra via pointwise operations.

• The thin category associated to the linear order $([0, \infty], \ge)$ of extended nonnegative real numbers with addition as the tensor product and internal hom as truncated subtraction.

• If $(M, \otimes, I)$ is any monoidal category, $I$ being the monoidal unit, the slice category $M/I$ inherits a monoidal product given by

$(X \stackrel{f}{\to} I) \otimes (Y \stackrel{g}{\to} I) = (X \otimes Y \stackrel{f \otimes g}{\to} I \otimes I \cong I)$

where the isomorphism displayed is the canonical one. This monoidal product is semicartesian. The forgetful functor $\Sigma: M/I \to M$ is strong monoidal, and is universal in the sense of exhibiting the fact that semicartesian monoidal functors and strong monoidal functors form a coreflective sub-bicategory of the bicategory of monoidal categories and strong monoidal functors. (Check this.)

• The category of nominal sets is cartesian closed but also semi-cartesian closed if taken with the separated product, see Chapter 3.4 of Pitts’ monograph Nominal Sets.

## Internal logic

The internal logic of a (symmetric) semicartesian monoidal category is affine logic, which is like linear logic but permits the weakening rule (and also the exchange rule, if the monoidal structure is symmetric).

## Properties

### Semicartesian vs. cartesian

In a semicartesian monoidal category, any tensor product of objects $x \otimes y$ comes equipped with morphisms

$p_x : x \otimes y \to x$
$p_y : x \otimes y \to y$

given by

$x \otimes y \stackrel{1 \otimes e_y}{\longrightarrow} x \otimes I \stackrel{r_x}{\longrightarrow} x$

and

$x \otimes y \stackrel{e_x \otimes 1}{\longrightarrow} I \otimes y \stackrel{\ell_y}{\longrightarrow} y$

respectively, where $e$ stands for the unique morphism to the terminal object and $r$, $\ell$ are the right and left unitors. We can thus ask whether $p_x$ and $p_y$ make $x \otimes y$ into the product of $x$ and $y$. If so, it is a theorem that $C$ is a cartesian monoidal category. (This theorem has been observed by Eilenberg and Kelly (1966, p.551), but they may not have been the first to note it.)

Alternatively, suppose that $(C, \otimes, I)$ is a symmetric monoidal category equipped with monoidal natural transformations $e_x : x \to I$ and $\Delta_x: x \to x \otimes x$ such that

$x \stackrel{\Delta_x}{\longrightarrow} x \otimes x \stackrel{e_x \otimes 1}{\longrightarrow} I \otimes x \stackrel{\ell_x}{\longrightarrow} x$

and

$x \stackrel{\Delta_x}{\longrightarrow} x \otimes x \stackrel{1 \otimes e_x}{\longrightarrow} x \otimes I \stackrel{r_x}{\longrightarrow} x$

are identity morphisms. Then $(C, \otimes, I)$ is a cartesian monoidal category (see this MO question for discussion of one of the technical details).

So, suppose $(C, \otimes, 1)$ is a semicartesian symmetric monoidal category. The unique map $e_x : x \to I$ is a monoidal natural transformation. Thus, if there exists a monoidal natural transformation $\Delta_x: x \to x \otimes x$ obeying the above two conditions, $(C, \otimes, 1)$ is cartesian. The converse is also true.

The characterization of cartesian monoidal categories in terms of $e$ and $\Delta$, apparently discovered by Robin Houston, is mentioned on page 47 of the slides at:

• John Baez, Universal algebra and diagrammatic reasoning, 2006. link

and as of 2014, Nick Gurski plans to write up the proof in a paper on semicartesian monads.

### Definition in terms of projections

For a monoidal category to be semicartesian it suffices that it admit a family of “projection morphisms”. Specifically, suppose $C$ is a monoidal category together with natural “projection” transformations $\pi^1_{X,Y}:X\otimes Y \to X$ such that $\pi^1_{I,I}:I\otimes I\to I$ is the unitor isomorphism. Then the composites $Y \cong I\otimes Y \to I$ form a cone under the identity functor with vertex $I$ whose component at $I$ is the identity; hence $I$ is a terminal object and so $C$ is semicartesian.

However, it doesn’t follow from this that the given projections $\pi^1_{X,Y}$ are the same as those derivable from semicartesianness! For that one needs extra axioms; see this cafe discussion for details.

### Colax functors

It is well-known that any functor between cartesian monoidal categories is automatically and uniquely colax monoidal; the colax structure maps are the comparison maps $F(x\times y) \to F x \times F y$ for the cartesian product. (This also follows from abstract nonsense given that the 2-monad for cartesian monoidal categories is colax-idempotent.) An inspection of the proof reveals that this property only requires the domain category to be semicartesian monoidal, although the codomain must still be cartesian.