category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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.
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 measurable locales, equivalently, the opposite category of commutative von Neumann algebras, equipped with the spatial tensor product, which corresponds to the measure-theoretic product of measurable spaces (but not the categorical product).
The category Cat with the non-cartesian funny tensor product.
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 $hom(x,y)$ 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 $hom(x,y)$ being the set of convex linear maps from $x$ to $y$, made into an barycentric algebra via pointwise operations.
The thin category associated to the linear order $([0, \infty], \ge)$ of extended nonnegative real numbers with addition as the tensor product and internal hom as truncated subtraction.
If $(M, \otimes, I)$ is any monoidal category, $I$ being the monoidal unit, the slice category $M/I$ inherits a monoidal product given by
where the isomorphism displayed is the canonical one. This monoidal product is semicartesian. The forgetful functor $\Sigma: M/I \to M$ is strong monoidal, and is universal in the sense of exhibiting the fact that semicartesian monoidal functors and strong monoidal functors form a coreflective sub-bicategory of the bicategory of monoidal categories and strong monoidal functors. (Check this.)
The category of nominal sets is cartesian closed but also semi-cartesian closed if taken with the separated product, see Chapter 3.4 of Pitts’ monograph Nominal Sets.
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).
In a semicartesian monoidal category, 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 has been observed by Eilenberg and Kelly (1966, p.551), but they may not have been the first to note it.)
Alternatively, suppose that $(C, \otimes, I)$ is a symmetric monoidal category equipped with monoidal natural transformations $e_x : x \to I$ and $\Delta_x: x \to x \otimes x$ such that
and
are identity morphisms. Then $(C, \otimes, I)$ is a cartesian monoidal category (see this MO question for discussion of one of the technical details).
So, suppose $(C, \otimes, 1)$ is a semicartesian symmetric monoidal category. The unique map $e_x : x \to I$ is a monoidal natural transformation. Thus, if there exists a monoidal natural transformation $\Delta_x: x \to x \otimes x$ obeying the above two conditions, $(C, \otimes, 1)$ is cartesian. The converse is also true.
The characterization of cartesian monoidal categories in terms of $e$ and $\Delta$, 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.
For a monoidal category to be semicartesian it suffices that it admit a family of “projection morphisms”. Specifically, suppose $C$ is a monoidal category together with natural “projection” transformations $\pi^1_{X,Y}:X\otimes Y \to X$ such that $\pi^1_{I,I}:I\otimes I\to I$ is the unitor isomorphism. Then the composites $Y \cong I\otimes Y \to I$ form a cone under the identity functor with vertex $I$ whose component at $I$ is the identity; hence $I$ is a terminal object and so $C$ is semicartesian.
However, it doesn’t follow from this that the given projections $\pi^1_{X,Y}$ are the same as those derivable from semicartesianness! For that one needs extra axioms; see this cafe discussion for details.
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 $F(x\times y) \to F x \times F y$ 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.
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)
A relevance monoidal category is the “dual” of a semicartesian monoidal category, with diagonals but not projections.
S. Eilenberg, M. G. Kelly, Closed Categories , pp.421-562 in Eilenberg et al. (eds.), Proceedings of the Conference on Categorical Algebra - La Jolla 1965 , Springer Heidelberg 1966.
Malte Gerhold?, Stephanie Lachs?, Michael Schürmann?, Categorial Lévy Processes, arXiv:1612.05139.
Tom Leinster, Monoidal Categories with Projections (blog discussion)
MO user ‘dremodaris’, A semicartesian monoidal category with diagonals is cartesian: proof? MO348480