Type of category in mathematics
In
mathematics, especially in
category theory, a closed monoidal category (or a monoidal closed category) is a
category that is both a
monoidal category and a
closed category in such a way that the structures are compatible.
A classic example is the
category of sets, Set, where the monoidal product of sets
and
is the usual
cartesian product
, and the
internal Hom
is the set of
functions from
to
. A non-
cartesian example is the
category of vector spaces, K-Vect, over a
field
. Here the monoidal product is the usual
tensor product of
vector spaces, and the internal Hom is the vector space of
linear maps from one vector space to another.
The
internal language of closed symmetric monoidal categories is
linear logic and the
type system is the
linear type system. Many examples of closed monoidal categories are
symmetric. However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulations of
linguistics; roughly speaking, this is because word-order in natural language matters.
Definition
A closed monoidal category is a
monoidal category
such that for every object
the
functor given by right tensoring with
![{\displaystyle A\mapsto A\otimes B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8266391df73af2a43237d11c2239059125caa996)
has a
right adjoint, written
![{\displaystyle A\mapsto (B\Rightarrow A).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd0c4a2f9f5bdc156d00f6a2627b57000a04e179)
This means that there exists a bijection, called '
currying', between the
Hom-sets
![{\displaystyle {\text{Hom}}_{\mathcal {C}}(A\otimes B,C)\cong {\text{Hom}}_{\mathcal {C}}(A,B\Rightarrow C)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6426fb56368a14db4b08fef3a819c3569a91fa10)
that is natural in both A and C. In a different, but common notation, one would say that the functor
![{\displaystyle -\otimes B:{\mathcal {C}}\to {\mathcal {C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60b10f0b8b9bc893ec5b7dc0e70f1fa4de9d7b1a)
has a right adjoint
![{\displaystyle [B,-]:{\mathcal {C}}\to {\mathcal {C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/479e1042d28f4ea82ab82253bb0c41e57c96d757)
Equivalently, a closed monoidal category
is a category equipped, for every two objects A and B, with
- an object
,
- a morphism
,
satisfying the following universal property: for every morphism
![{\displaystyle f:X\otimes A\to B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9580990ea7a6f774cb9199fe30f6ce5a9c08a5c0)
there exists a unique morphism
![{\displaystyle h:X\to A\Rightarrow B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8da9b5e7799c0d9a43f041f7006b62235e5c021)
such that
![{\displaystyle f=\mathrm {eval} _{A,B}\circ (h\otimes \mathrm {id} _{A}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc38e233930695fa829d3a5c31fba531de47e8ba)
It can be shown[
citation needed] that this construction defines a functor
. This functor is called the
internal Hom functor, and the object
is called the internal Hom of
and
. Many other notations are in common use for the internal Hom. When the tensor product on
is the cartesian product, the usual notation is
and this object is called the
exponential object.
Biclosed and symmetric categories
Strictly speaking, we have defined a right closed monoidal category, since we required that right tensoring with any object
has a right adjoint. In a left closed monoidal category, we instead demand that the functor of left tensoring with any object
![{\displaystyle B\mapsto A\otimes B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e0dce297f888d00d62c4ba4dd28e86409699b9b)
have a right adjoint
![{\displaystyle B\mapsto (B\Leftarrow A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29ca402cf07798317006db8299d70bbc649ee540)
A biclosed monoidal category is a monoidal category that is both left and right closed.
A
symmetric monoidal category is left closed if and only if it is right closed. Thus we may safely speak of a 'symmetric monoidal closed category' without specifying whether it is left or right closed. In fact, the same is true more generally for
braided monoidal categories: since the braiding makes
naturally isomorphic to
, the distinction between tensoring on the left and tensoring on the right becomes immaterial, so every right closed braided monoidal category becomes left closed in a canonical way, and vice versa.
We have described closed monoidal categories as monoidal categories with an extra property. One can equivalently define a closed monoidal category to be a
closed category with an extra property. Namely, we can demand the existence of a
tensor product that is
left adjoint to the
internal Hom functor.
In this approach, closed monoidal categories are also called monoidal closed categories.[
citation needed]
Examples
- Every
cartesian closed category is a symmetric, monoidal closed category, when the monoidal structure is the cartesian product structure. The internal Hom functor is given by the
exponential object
.
- In particular, the
category of sets, Set, is a symmetric, closed monoidal category. Here the internal Hom
is just the set of functions from
to
.
- The
category of modules, R-Mod over a
commutative ring R is a non-cartesian, symmetric, monoidal closed category. The monoidal product is given by the
tensor product of modules and the internal Hom
is given by the space of
R-linear maps
with its natural R-module structure.
- In particular, the category of vector spaces over a field
is a symmetric, closed monoidal category.
-
Abelian groups can be regarded as Z-modules, so the
category of abelian groups is also a symmetric, closed monoidal category.
- A
symmetric compact closed category is a symmetric monoidal closed category in which the internal Hom functor
is given by
. The canonical example is the category of finite-dimensional vector spaces, FdVect.
Counterexamples
- The
category of rings is a symmetric, monoidal category under the
tensor product of rings, with
serving as the unit object. This category is not closed. If it were, there would be exactly one homomorphism between any pair of rings:
. The same holds for the category of R-
algebras over a
commutative ring R.
See also
References