With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
A cartesian monoidal category (usually just called a cartesian category), is a monoidal category whose monoidal structure is given by the category-theoretic product (and so whose unit is a terminal object).
A cartesian monoidal category which is also closed is called a cartesian closed category.
A strong monoidal functor between cartesian categories is called a cartesian functor.
Any category with finite products can be considered as a cartesian monoidal category (as long as we have either (1) a specified product for each pair of objects, (2) a global axiom of choice, or (3) we allow the monoidal product to be an anafunctor).
The term cartesian category usually means a category with finite products but can also mean a finitely complete category, so we avoid that term.
Among general monoidal categories, the cartesian monoidal categories have a number of special and important properties, such as the existence of diagonal maps $\Delta_x : x \to x\otimes x$ and augmentations $e_x: x \to I$ for any object $x$, cf. the structural rules in the corresponding internal type theory: In applications to computer science we can think of $\Delta$ as ‘duplicating data’ and $e$ as ‘deleting’ data. These maps make any object into a comonoid: (see also at comonoid – In a cartesian monoidal category):
(comonoid objects)
In fact, any object in a cartesian monoidal category becomes a comonoid in a unique way, and this is automatically a cocommutative comonoid structure. Furthermore, this comonoid structure is ‘natural’, meaning that the all morphisms between objects preserve their comonoid structure. In other words, any morphism $f: x \to y$ is a comonoid homomorphism:
Moreover, one can show (e.g. Fox 1976 or Heunen-Vicary 2012, p. 79 (p. 85 of the pdf)) that any symmetric monoidal category equipped with suitably well-behaved diagonal and augmentation maps must in fact be cartesian monoidal. More precisely: suppose $C$ is a symmetric monoidal category equipped with monoidal natural transformations
and
such that the following composites are identity morphisms:
where $r$, $\ell$ are the right and left unitors. Then $C$ is cartesian.
Heuristically: a symmetric monoidal category is cartesian if we can duplicate and delete data, and ‘duplicating a piece of data and then deleting one copy is the same as not doing anything’.
A related theorem describes cartesian monoidal categories as monoidal categories satisfying two properties involving the unit object. First, we say a monoidal category $C$ is semicartesian if the unit for the tensor product is terminal. If this is true, any tensor product of objects $x \otimes y$ comes equipped with morphisms
given by
and
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.)
Cartesian categories may be freely generated from sets, categories, signatures, etc., as explained at free cartesian category.
As outlined above, cartesianness is an algebraic structure on top of a symmetric monoidal structure. This means that the 2-category of cartesian monoidal categories $\mathbf{CartMonCat}$ is monadic over $\mathbf{SymMonCat}$, i.e. there is a free-forgetful adjunction
Thomas Fox showed in (Fox 1976) that $\mathbf{Cart}$ is also comonadic, that is, that $U$ admits a right adjoint $C \dashv U$, given by constructing the category of cocommutative comonoids in a given symmetric monoidal category $\mathbf{M}$.
The characterization of cartesian monoidal categories as symmetric monoidal categories with extra structure:
Discussion with an eye towards finite quantum mechanics in terms of dagger-compact categories is in
Chris Heunen, Jamie Vicary, Lectures on categorical quantum mechanics, 2012 (pdf)
Kosta Došen and Zoran Petrić. The maximality of cartesian categories. Mathematical Logic Quarterly: Mathematical Logic Quarterly 47.1 (2001): 137-144.
Last revised on August 16, 2024 at 06:55:04. See the history of this page for a list of all contributions to it.