nLab tensorial strength




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


Given a monoidal category VV, a tensorial left-strength for a functor

F:VVF : V \to V

is usually defined as a natural transformation

β v,w:vF(w)F(vw) \beta_{v, w} \;\colon\; v \otimes F(w) \longrightarrow F(v \otimes w)

making the following diagrams commute:

(uv)F(w) β uv,w F((uv)w) α u,v,F(w) F(α u,v,w) u(vF(w)) 1 uβ v,w uF(vw) β u,vw F(u(vw)) \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)) }


IF(v) β I,v F(Iv) λ F(v) F(λ v) F(v) \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 α\alpha is the associator and λ\lambda is the unitor.

A tensorial right-strength (not to be confused with a tensorial costrength) is defined symmetrically, as a natural transformation

γ v,w:F(w)vF(wv)\gamma_{v, w}: F(w) \otimes v \to F(w \otimes v)

When VV 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 VV, but for functors between any categories that are “tensored over VV.”

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 VActV-Act where:

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

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

A bit more precisely:

VAct=Lax(BV,Cat)V-Act = Lax(B V, Cat)

Here BVB V is VV regarded as a 1-object weak 2-category, CatCat is regarded as a 2-category, and LaxLax is the bicategory of

There is a forgetful 2-functor

VActCatV-Act \to Cat

and a tensorial strength for a functor F:VVF: V \to V is a way of lifting it to a morphism F˜:V˜V˜\tilde{F}: \tilde{V} \to \tilde{V} in VActV-Act, where V˜\tilde{V} is VV 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 T:VVT: V \to V. (The concept doesn’t make much immediate sense if TT 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=SetV = Set, they’re sort of invisible. Every functor T:SetSetT: 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 VV is closed monoidal. Here VV is enriched in itself, and one of the most important aspects of strengths is this:

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

Here’s how that works. An enrichment on TT in VV-enriched category theory consists of a natural family of morphisms in VV,

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

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

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

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

T(hom(a,b)a)T(eval a,b)T(b)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 TT 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=SetV = Set are so “invisible” – every functor T:SetSetT: Set \to Set is SetSet-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]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]B[X]T: B[X] \to B[X]

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


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

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.

Last revised on May 3, 2023 at 10:59:27. See the history of this page for a list of all contributions to it.