nLab
semicartesian monoidal category

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

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.

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 xyx \otimes y comes equipped with morphisms

p x:xyx p_x : x \otimes y \to x
p y:xyy p_y : x \otimes y \to y

given by

xy1e yxIr xx x \otimes y \stackrel{1 \otimes e_y}{\longrightarrow} x \otimes I \stackrel{r_x}{\longrightarrow} x

and

xye x1Iy yy x \otimes y \stackrel{e_x \otimes 1}{\longrightarrow} I \otimes y \stackrel{\ell_y}{\longrightarrow} y

respectively, where ee stands for the unique morphism to the terminal object and rr, \ell are the right and left unitors. We can thus ask whether p xp_x and p yp_y make xyx \otimes y into the product of xx and yy. If so, it is a theorem that CC 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,,I)(C, \otimes, I) is a symmetric monoidal category equipped with monoidal natural transformations e x:xIe_x : x \to I and Δ x:xxx\Delta_x: x \to x \otimes x such that

xΔ xxxe x1Ix xx x \stackrel{\Delta_x}{\longrightarrow} x \otimes x \stackrel{e_x \otimes 1}{\longrightarrow} I \otimes x \stackrel{\ell_x}{\longrightarrow} x

and

xΔ xxx1e xxIr xx 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,,I)(C, \otimes, I) is a cartesian monoidal category (see this MO question for discussion of one of the technical details).

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

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

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 CC is a monoidal category together with natural “projection” transformations π X,Y 1:XYX\pi^1_{X,Y}:X\otimes Y \to X such that π I,I 1:III\pi^1_{I,I}:I\otimes I\to I is the unitor isomorphism. Then the composites YIYIY \cong I\otimes Y \to I form a cone under the identity functor with vertex II whose component at II is the identity; hence II is a terminal object and so CC is semicartesian.

However, it doesn’t follow from this that the given projections π X,Y 1\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×y)Fx×FyF(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:

Reference