nLab semicartesian monoidal category

Definition

A monoidal category is semicartesian if the unit for the tensor product is a terminal object. This is a weaker version of the concept of cartesian monoidal category (where moreover the tensor product is required to be the cartesian product).

Dually, a monoidal category is semicocartesian if the unit for the tensor product is an initial object. This is a weaker version of the concept of cocartesian monoidal category (where in addition the tensor product is required to be the coproduct.)

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 opposite of the category FinInj of finite sets with injections as morphisms with disjoint union as tensor product. This is the free symmetric semicartesian monoidal category generated by one object.

• Similarly, the opposite of the category of finite, ordered sets with injective order-preserving functions as morphisms and disjoint union of orders as tensor is the free semicartesian monoidal category generated by one object.

• The category of nominal sets, equivalently the Schanuel topos has a semi-cartesian closed structure with separated product, see Chapter 3.4 of Pitts’ monograph Nominal Sets. This is related to the first example as the Schanuel topos is a topos of sheaves over FinInj, and the semicartesian monoidal structure is given by Day convolution of sheaves using the semicartesian monoidal structure of FinInj.

• 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.)

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 $\Delta$ is mentioned on page 47 of the slides at:

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

However, the characterisation was certainly known earlier: for instance, it appears in the thesis of Kegelmann, in which it is called folklore.

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.

Semicartesian operads

The notion of semicartesian operad? is a type of generalized multicategory which corresponds to semicartesian monoidal categories in the same way that operads correspond to (perhaps symmetric) monoidal categories and Lawvere theories correspond to cartesian monoidal categories. Applications of semicartesian operads include:

• Generalized operads in classical algebraic topology (blog post) – this also uses the above fact about colax functors

• Characterizing finite measure spaces (blog comment)

Related pages

• cartesian monoidal category

• A relevance monoidal category is the “dual” of a semicartesian monoidal category, with diagonals but not projections.

• category of operators

Reference

• Samuel Eilenberg, G. Max Kelly, Closed Categories, pp. 421-562 in: S. Eilenberg, D. K. Harrison, S. MacLane, H. Röhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) [doi:10.1007/978-3-642-99902-4]

• Malte Gerhold?, Stephanie Lachs?, Michael Schürmann?, Categorial Lévy Processes, arXiv:1612.05139.

• Tom Leinster, Monoidal Categories with Projections (blog discussion)

• MO user ‘dremodaris’, A semicartesian monoidal category with diagonals is cartesian: proof? MO348480

• Mathias Kegelmann, Continuous domains in logical form, Electronic Notes in Theoretical Computer Science 49 (2002): 1-166.