nLab tensorial strength

Idea

In the simplest case, the notion of tensorial strength is a way to talk about when an endofunctor $T : V \to V$ of a closed monoidal category is a $V$-enriched functor when $V$ is regarded as being a $V$-enriched category. But in fact the concept makes sense even when $V$ is not closed.

Definition

Given a monoidal category $V$, a tensorial left-strength for a functor

$$F : V \to V$$

is usually defined as a natural transformation

$$\beta_{v, w} \;\colon\; v \otimes F(w) \longrightarrow F(v \otimes w)$$

making the following diagrams commute:

$$\array{ (u \otimes v) \otimes F(w) & & \overset{\beta_{u\otimes v,w}}{\to} & & F((u\otimes v)\otimes w)\\ ^\mathllap{\alpha_{u,v,F(w)}}\downarrow & & & & \downarrow^{\mathrlap{F(\alpha_{u,v,w})}}\\ u\otimes (v\otimes F(w)) & \underset{1_u\otimes\beta_{v,w}}{\to} & u\otimes F(v\otimes w) & \underset{\beta_{u,v\otimes w}}{\to} & F(u\otimes (v\otimes w)) }$$

and

$$\array{ I\otimes F(v) & \overset{\beta_{I,v}}{\to} & F(I\otimes v) \\ & _\mathllap{\lambda_{F(v)}}\searrow & \downarrow^\mathrlap{F(\lambda_v)}\\ & & F(v) }$$

where $\alpha$ is the associator and $\lambda$ is the unitor.

A tensorial right-strength is defined symmetrically, as a natural transformation

$$\gamma_{v, w}: F(w) \otimes v \to F(w \otimes v)$$

When $V$ is symmetric, tensorial left strengths are equivalently tensorial right strengths, and in this setting, one usually drops “left-” or “right-” and simply talks about tensorial strengths.

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 $V$, but for functors between any categories that are “tensored over $V$.”

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 $V-Act$ where:

• the objects are categories $C$ equipped with a lax left $V$-action, i.e. a lax monoidal functor $\alpha : V \to End(C)$,
• the morphisms are functors laxly preserving this action, and
• the 2-morphisms are natural transformations compatible with this action.

$V$ has a canonical left action on itself, which allows us to think of it as an object of $V-Act$. So, given a functor $F: V \to V$, a tensorial strength for $F$ is defined to be a way of equipping it with the structure of a morphism in $V-Act$.

A bit more precisely:

$$V-Act = Lax(B V, Cat)$$

Here $B V$ is $V$ regarded as a 1-object weak 2-category, $Cat$ is regarded as a 2-category, and $Lax$ is the bicategory of

• weak 2-categories
• lax functors
• lax natural transformations

There is a forgetful 2-functor

$$V-Act \to Cat$$

and a tensorial strength for a functor $F: V \to V$ is a way of lifting it to a morphism $\tilde{F}: \tilde{V} \to \tilde{V}$ in $V-Act$, where $\tilde{V}$ is $V$ equipped with its canonical left action on itself.

Description

The first thing to notice about (covariant) tensorial strengths is that they attach to a functor from a monoidal category to itself, say $T: V \to V$. (The concept doesn’t make much immediate sense if $T$ 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 $V = Set$, they’re sort of invisible. Every functor $T: Set \to Set$ has one, in fact a canonical one! This will make more sense in a moment.

Strengths are easier to understand by considering the case where $V$ is closed monoidal. Here $V$ is enriched in itself, and one of the most important aspects of strengths is this:

• A functor $T: V \to V$ with a strength is the same thing as a $V$-enrichment on $T$.

Here’s how that works. An enrichment on $T$ in $V$-enriched category theory consists of a natural family of morphisms in $V$,

$$hom(a, b) \to hom(T(a), T(b)),$$

satisfying some suitable axioms. Here $hom$ denotes the enrichment of $V$ in itself – the internal hom. Starting from a strength on $T$, you can cook up an enrichment on $T$, roughly as follows: the map above is equivalent by hom-tensor adjunction to a map

$$hom(a, b) \otimes T(a) \to T(b)$$

and now the strength allows you to slide $hom(a, b)$ inside $T$, and from there you just apply $T$ to the evaluation:

$$T(hom(a, b) \otimes a) \overset{T(eval_{a, b})}{\to} T(b)$$

and presto, you’re done. The strength axioms ensure that this enrichment structure on $T$ 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 $V = Set$ are so “invisible” – every functor $T: Set \to Set$ is $Set$-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, $B[X]$. Then again we can think of this, or of any Heyting algebra, as enriched in itself. Further, we have definable unary operators

$$T: B[X] \to B[X]$$

in the theory, such as $T = p \wedge (-)$, or $T = p \Rightarrow (-)$, 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.)

Related concepts

• cartesian closed functor
• strong monad

References

The tentative ‘more conceptual’ definition of tensorial strength, as well as the ‘description’ above, arose in this discusssion:

• John Baez, The Monads Hurt My Head – But Not Anymore (blog)

The concept of tensorial strength is a prerequisite for the concept of strong monad, so see the further discussion there. An original reference is

• Anders Kock, Strong functors and monoidal monads, Arch. Math. (Basel) 23 (1972), 113–120.