question about monoidal categories
Let F be a functor from a monoidal category M to a category S. We are given beta(p,a) : F(p) --> F(p*a) natural in p,a in M. If I tell you that, in addition to naturality, beta is "monoidal", I'm sure you will immediately guess what I mean by this, viz. (a) for any p,a,b in M beta(p,a) ; beta(p*a,b) = beta(p,a*b) ; F(alpha(p,a,b)) (b) for any p in M beta(p,1) = F(rho(p)) Yet I cannot see any reason for giving the name "monoidality" to (a)-(b). It doesn't appear to be a monoidal natural transformation in the official sense. There are no monoidal functors in sight. Can somebody please justify my usage? Paul
Dear Paul, Here is one possible answer to your question. One has a notion of action of a monoidal category V on an arbitrary category X, which generalises that of action by a monoid on a set; thus we have a functor (-) * (-): X x V --> X together with natural isomorphisms x * I ~ x and x * (v * w) ~ (x * v) * w satisfying pentagon and triangle axioms. Formally, one may define an action of V on X to be a strong monoidal functor V --> [X, X], where the latter is equipped with its compositional monoidal structure. If we are given two categories X and Y equipped with an action by V, then we have a notion of equivariant morphism between them; namely a functor F: X --> Y together with natural morphisms m_{x, v} : F(x) * v --> F(x * v) obeying axioms like those for a monoidal functor. This is what one might call a lax equivariant morphism; if the m_{x, v}'s are all invertible we should rather call it strong, whilst if they point in the opposite direction then what we have is an oplax morphism. The situation you have described is a special case of an lax equivariant morphism. You have a monoidal category M, and a functor F : M --> S. Now, M has a canonical action on itself induced by tensoring on the right (the "right regular representation"); and it has a trivial action on S given by s * m = s for all s and m. Your natural transformation beta can now be written as beta(p, a) : F(p) * a --> F(p * a), and your two axioms are precisely the axioms required for beta to equip F with the structure of a lax equivariant morphism. This whole area of monoidal actions is slightly folklorish but a useful source is: George Janelidze and Max Kelly, "A note on actions of a monoidal category", TAC Vol. 9, No. 4 Also worth mentioning is the work of Paddy McCrudden who has studied actions by a symmetric monoidal V under the name "V-actegories". Hope this is of some help, Richard --On 07 February 2008 20:05 Paul B Levy wrote:
Let F be a functor from a monoidal category M to a category S.
We are given
beta(p,a) : F(p) --> F(p*a)
natural in p,a in M.
If I tell you that, in addition to naturality, beta is "monoidal", I'm sure you will immediately guess what I mean by this, viz.
(a) for any p,a,b in M
beta(p,a) ; beta(p*a,b) = beta(p,a*b) ; F(alpha(p,a,b))
(b) for any p in M
beta(p,1) = F(rho(p))
Yet I cannot see any reason for giving the name "monoidality" to (a)-(b).
It doesn't appear to be a monoidal natural transformation in the official sense. There are no monoidal functors in sight.
Can somebody please justify my usage?
Paul
Hi Paul, I think that I have written previously to the list about the possibility of a monoidal functor acting on a mere functor, and what you have is an instance of this notion. Here the monoidal functor is the unique functor M ---> T, where T is the terminal (monoidal) category. Your beta is a right action of this guy on F. In general, a right action of monoidal A --U--> C on mere P --F--> S requires a right action of A on P and a right action of C on S as well as a natural transformation F(p)*U(a) --beta(p,a)--> F(p*a) satisfying the appropriate associativity and unit axioms. In your case S is equipped with the trivial right T-action (x*1=x), and M with its canonical right M-action (a*b=a*b). The axioms are identical. Cheers, Jeff. --- Paul B Levy <P.B.Levy@cs.bham.ac.uk> wrote:
Let F be a functor from a monoidal category M to a category S.
We are given
beta(p,a) : F(p) --> F(p*a)
natural in p,a in M.
If I tell you that, in addition to naturality, beta is "monoidal", I'm sure you will immediately guess what I mean by this, viz.
(a) for any p,a,b in M
beta(p,a) ; beta(p*a,b) = beta(p,a*b) ; F(alpha(p,a,b))
(b) for any p in M
beta(p,1) = F(rho(p))
Yet I cannot see any reason for giving the name "monoidality" to (a)-(b).
It doesn't appear to be a monoidal natural transformation in the official sense. There are no monoidal functors in sight.
Can somebody please justify my usage?
Paul
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Dear Paul, Let us forget about the monoidal unit and fix (a). You define a strict semi? monoidal structure on S letting x*y = x. Then your pair (F, beta) is a lax monoidal functor M --> S. Your condition (a) is the hexagon of consistency with alpha, which here reduces to: F(p) = F(p) || | F(p) F(p*a) | | F(p*(a*b) --> F((p*a)*b) (a single | stands for a downward beta) Now, to fix also (b), I guess you should add to S a new object which is a strict identity for the tensor and work out things. However, if your problem is only about terminology and you do not want to use the tensor on S in the sequel (eg to compose F with other monoidal functors), you might not bother about that. Best regards Marco
Hello, here is an answer to Paul's question: The data (F,beta) defines a (lax?) functor between "monoidal categories with right units but not left units" [henceforth MR-categories]. Every pointed category can be considered as an MR-category, in fact a strict one. The tensor product is left projection. The right unit is the point of the category (although every object of the category behaves like a right unit for this category.) In particular, the category S in Paul's email, with point F(1), can be seen as an MR-category. Of course, every monoidal category, including M, is an MR-category too. The data for a lax MR-functor M->S consists of a functor F:M->S, a morphism F(i)->F(i), which we can take as identity, and a natural transformation F(p)*F(a) --> F(p*a) which in this case amounts to a map beta:F(p) --> F(p*a) A monoidal functor must satisfy three coherence conditions, for associativity, left identity and right identity. For an MR-functor, there are only axioms for associativity and right identity, and these are exactly the axioms that Paul gave. Hope that makes sense! All the best, Sam. On 7 Feb 2008, at 20:05, Paul B Levy wrote:
Let F be a functor from a monoidal category M to a category S.
We are given
beta(p,a) : F(p) --> F(p*a)
natural in p,a in M.
If I tell you that, in addition to naturality, beta is "monoidal", I'm sure you will immediately guess what I mean by this, viz.
(a) for any p,a,b in M
beta(p,a) ; beta(p*a,b) = beta(p,a*b) ; F(alpha(p,a,b))
(b) for any p in M
beta(p,1) = F(rho(p))
Yet I cannot see any reason for giving the name "monoidality" to (a)-(b).
It doesn't appear to be a monoidal natural transformation in the official sense. There are no monoidal functors in sight.
Can somebody please justify my usage?
Paul
As an addendum to the interesting answers that have been given so far to Paul's question, it is perhaps worth pointing out that using an old result of Max Kelly's, the situation Paul describes can be expressed purely in terms of a strong monoidal functor. Given a monoidal category M and functor F : M --> S as in Paul's message, we define a strict monoidal category {F, F} as follows. Its objects are triples (G, H, a) fitting into a diagram of functors and natural transformations G M -----> M | | | a | F | => | F | | v v S -----> S H Its morphisms (G, H, a) --> (G', H', a') are pairs of natural transformations b : G => G', c : H => H' satisfying the obvious compatibility condition with a and a'. The tensor product is given by pasting squares next to each other horizontally. There are strict monoidal functors p_1 : {F, F} -> [M, M] and p_2 : {F, F} -> [S, S] sending (G, H, a) to G and H respectively; and as in my previous message we have strong monoidal functors R : M ---> [M, M] m |--> (-) * m T : M ---> [S, S] m |--> id_S corresponding to the right regular action of M on itself, and to the trivial action of M on S. Now to give the natural transformation beta of Paul's message, satisfying his "monoidality" conditions, is precisely to give a strong monoidal functor B: M --> {F, F} rendering the diagram _ {F, F} .| | . | B . | (p_1, p_2) . | . v M --------> [M, M] x [S, S] (R, T) commutative. I believe this technique originates in the paper "Coherence theorems for lax algebras and for distributive laws", G.M. Kelly, LNM 420. A good place to learn more about it is in Section 2 of "On property-like structures", G.M. Kelly and S. Lack, TAC Vol. 3, No. 9 Richard --On 07 February 2008 20:05 Paul B Levy wrote:
Let F be a functor from a monoidal category M to a category S.
We are given
beta(p,a) : F(p) --> F(p*a)
natural in p,a in M.
If I tell you that, in addition to naturality, beta is "monoidal", I'm sure you will immediately guess what I mean by this, viz.
(a) for any p,a,b in M
beta(p,a) ; beta(p*a,b) = beta(p,a*b) ; F(alpha(p,a,b))
(b) for any p in M
beta(p,1) = F(rho(p))
Yet I cannot see any reason for giving the name "monoidality" to (a)-(b).
It doesn't appear to be a monoidal natural transformation in the official sense. There are no monoidal functors in sight.
Can somebody please justify my usage?
Paul
participants (5)
-
Jeff Egger -
Marco Grandis -
Paul B Levy -
Richard Garner -
Sam Staton