nLab
semicartesian monoidal category

Definition

A monoidal category is semicartesian if the unit for the tensor product is a terminal object. This a weakening of the concept of cartesian monoidal category, which might seem like pointless centipede mathematics were it not for the existence of interesting examples and applications.

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 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 $\mathrm{hom}(x,y)$ 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 $\mathrm{hom}(x,y)$ being the set of convex linear maps from $x$ to $y$, made into an barycentric algebra via pointwise operations.

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_x : x \otimes y \to x

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

given by

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

and

$$x\otimes y\stackrel{e_x\otimes 1}{⟶}I\otimes y\stackrel{{\ell }_y}{⟶}y$$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 is probably in Eilenberg and Kelly’s paper on closed categories, but they may not have been the first to note it.) This also follows if we posit the existence of a natural diagonal morphism $x\to x\otimes x$.

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 $F(x\times y)\to Fx\times Fy$ 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)