In the simplest case, the notion of tensorial strength is a way to talk about when an endofunctor of a closed monoidal category is a -enriched functor when is regarded as being a -enriched category. But in fact the concept makes sense even when is not closed.
Given a monoidal category , a tensorial strength for a functor
is usually defined as a natural transformation
making two diagrams commute:
A functor equipped with a tensorial strength is called a strong functor (not to be confused with a strong 2-functor, which is another name for a pseudofunctor, i.e. a lax 2-functor whose coherence cells are invertible).
More generally, the notion makes sense not just for endofunctors of , but for functors between any categories that are “tensored over .”
Tentative – The following reformulation of the standard definition is tentative. See the discussion referenced below
It seems we may phrase this definition more conceptually as follows. There is a bicategory where:
has a canonical left action on itself, which allows us to think of it as an object of . So, given a functor , a tensorial strength for is defined to be a way of equipping it with the structure of a morphism in .
A bit more precisely:
Here is regarded as a 1-object weak 2-category, is regarded as a 2-category, and is the bicategory of
There is a forgetful 2-functor
and a tensorial strength for a functor is a way of lifting it to a morphism in , where is equipped with its canonical left action on itself.
The first thing to notice about (covariant) tensorial strengths is that they attach to a functor from a monoidal category to itself, say . (The concept doesn’t make much immediate sense if is a functor between different monoidal categories.)
One possible reason why it may be hard to grasp the notion of strength is because in the case , they’re sort of invisible. Every functor has one, in fact a canonical one! This will make more sense in a moment.
Here’s how that works. An enrichment on in -enriched category theory consists of a natural family of morphisms in ,
satisfying some suitable axioms. Here denotes the enrichment of in itself – the internal hom. Starting from a strength on , you can cook up an enrichment on , roughly as follows: the map above is equivalent by hom-tensor adjunction to a map
and now the strength allows you to slide inside , and from there you just apply to the evaluation:
and presto, you’re done. The strength axioms ensure that this enrichment structure on satisfies the axioms you need for a functor to be enriched.
(And going back the other way, from an enrichment to a strength, is also easy – there you have to dualize. That is, instead of using the evaluation which is the counit of the hom-tensor adjunction, you use the coevaluation which is the unit. )
After some work, one can convince oneself that the notions of strength and enrichment for endofunctors on closed monoidal categories really are equivalent notions.
And now we can understand why strengths in the case are so “invisible” – every functor is -enriched; that’s what we mean by an ordinary functor!
There’s rather a lot more one could say about strengths, and I may come back to more of that later, but I would like to say that strengths are kind of a trade secret. The first mathematician I know of who intuitively grasped strength was C.S. Peirce! And particularly in his Alpha graphs, the notion of strength plays an important role.
The insight here can be related back to the enrichment = strength phenomenon. Suppose for instance we’re in the theory of Boolean algebras – say we’re studying the structure of a free Boolean algebra on a set of generators, . Then again we can think of this, or of any Heyting algebra, as enriched in itself. Further, we have definable unary operators
in the theory, such as , or , etc. The great discovery of Peirce is that any definable unary operator in the theory of Boolean algebras carries a strength, or if you prefer is enriched. That may be taken to be the essence of Peirce’s iteration rule for Alpha, and it categorifies right over to a similar statement for the theory of closed categories, star-autonomous categories, what have you: all definable unary covariant functors in such theories carry canonical strengths.
(Peirce went a little further, and incorporated notions of contravariant strength as well.)
The tentative ‘more conceptual’ definition of tensorial strength, as well as the ‘description’ above, arose in this discusssion:
The concept of tensorial strength is a prerequisite for the concept of strong monad, so see the further discussion there. An original reference is