category with duals (list of them)
dualizable object (what they have)
The term tensor product has many different but closely related meanings.
In its original sense a tensor product is a representing object for a suitable sort of bilinear map and multilinear map. The most classical versions are for vector spaces (modules over a field), and more generally modules over a ring. In modern language this takes place in a multicategory.
Consequently, the functor which is part of the data of any monoidal category is also often called a tensor product, since in many examples of monoidal categories it is induced from a tensor product in the above sense (and in fact, any monoidal category underlies a multicategory in a canonical way). In parts of the literature (certain) abelian monoidal categories are even addressed as tensor categories.
Given two objects in a monoidal category with a right and left action, respectively, of some monoid , their tensor product over is the quotient of their tensor product in by this action. If is commutative, then this is a special case of the tensor product in a multicategory.
This generalizes to modules over monads in a bicategory, which includes the notion of tensor product of functors.
Finally, tensor products in a multicategory and tensor products over monads in a bicategory are both special cases of tensor products in an virtual double category.
For a multicategory and and objects in , the tensor product is defined to be an object equipped with a universal multimorphism in that any multimorphism factors uniquely through via a (1-ary) morphism .
is the category Ab of abelian groups, made into a multicategory using multilinear maps as the multimorphisms, then we get the usual tensor product of abelian groups.? That is, is equipped with a universal map from (as a set) to such that this map is linear (a group homomorphism) in each argument separately. This tensor product can also be constructed explicitly by
Note that in this case, is not a subobject or a quotient of the cartesian product . However, in many other cases the tensor product in a multicategory can be obtained as a quotient of some other pre-existing product; see tensor product of modules below.
Other examples of tensor products in multicategories:
In particular, any closed category (even if not monoidal) has an underlying multicategory. Tensor products in this multicategory are characterized by the adjointness relation
This may be the oldest notion of tensor product, since the definition of the internal-hom of abelian groups and vector spaces, unlike that of their tensor product, is intuitively obvious.
While the universal property referred to above (every bilinear map factors uniquely through via a map ) suffices to define the tensor product, it does not suffice to prove that it is associative and unital. For this we need the stronger property that any multilinear map factors uniquely through via a multilinear map .
Let be a commutative ring and consider the multicategory Mod of -modules and -multilinear maps. In this case the tensor product of modules of -modules and can be constructed as the quotient of the tensor product of abelian groups underlying them by the action of ; that is,
More category-theoretically, this can be constructed as the coequalizer of the two maps
given by the action of on and on .
If is a field, then -modules are vector spaces; this gives probably the most familiar case of a tensor product spaces, which is also probably the situation where the concept was first conceived.
This tensor product can be generalized to the case when is not commutative, as long as is a right -module and is a left -module. More generally yet, if is a monoid in any monoidal category (a ring being a monoid in Ab with its tensor product), we can define the tensor product of a left and a right -module in an analogous way. If is a commutative monoid in a symmetric monoidal category, so that left and right -modules coincide, then is again an -module, while if is not commutative then will no longer be an -module of any sort.
Not all tensor products in multicategories are instances of this construction. In particular, the tensor product in Ab is not the tensor product of modules over any monoid in the cartesian monoidal category Set. Abelian groups can be considered as “sets with an action by something,” but that something is more complicated than a monoid: it is a special sort of monad called a commutative theory.
Conversely, if is a commutative monoid in a symmetric monoidal category, there is a multicategory of -modules whose tensor product agrees with the coequalizer defined above, but if is not commutative this is impossible. However, see the section on tensor products in virtual double categories, below.
The tensor product of left and right modules over a noncommutative monoid in a monoidal category is a special case of the tensor product of modules for a monad in a bicategory. If is a monad in a bicategory , a right -module is a 1-cell with an action by , a left -module is a 1-cell with an action by , and their tensor product, if it exists, is a 1-cell given by a similar coequalizer. Regarding a monoidal category as a 1-object bicategory, this recovers the above definition.
For example, consider the bicategory of -valued matrices for some monoidal category . A monad in is a -enriched category , an -bimodule is a functor , an -bimodule is a functor , and their tensor product in is a classical construction called the tensor product of functors. It can also be defined as a coend.
A virtual double category is a common generalization of a multicategory and a bicategory (and actually of a double category). Among other things, it has objects, 1-cells, and “multi-2-cells.” We leave it to the reader to define a notion of tensor product of 1-cells in such a context, analogous to the tensor product of objects in a multicategory. A multicategory can be regarded as a 1-object virtual double category, so this generalizes the notion of tensor product in a multicategory.
On the other hand, in any bicategory (in fact, any double category) there is a virtual double category whose objects are monads and whose 1-cells are bimodules, and the tensor product in this virtual double category is the tensor product of modules in a bicategory defined above. Thus, tensor products in a virtual double category include all notions of tensor product discussed above.
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Exposition of the tensor product of -modules is for instance in