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.) This also follows if we posit the existence of a natural diagonal morphism xxxx \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×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 May 20, 2011 21:52:24 by Mike Shulman (