semicartesian monoidal category


Monoidal categories

Category theory

Semicartesian monoidal categories


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.


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


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


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 is probably in Eilenberg and Kelly’s paper on closed categories, but they may not have been the first to note it.)

Alternatively, suppose that (C,,I)(C, \otimes, I) is a 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


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.

So, suppose (C,,1)(C, \otimes, 1) is a semicartesian 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 here:

  • John Baez, Universal algebra and diagrammatic reasoning, 2006. [[pdf](]

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

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:

Revised on January 20, 2015 20:39:02 by John Baez (