question about monoidal categories

question about monoidal categories

[claudio pisani]

Dear categorists,

suppose V is a monoidal category, with underlying category V_0, and X is an ordinary category. Then (if I am not mistaken) the functor category V_0^X has a monoidal structure, defined point-wise by that  of V and each functor f:X->Y gives a strong monoidal functor.

First question : supposing V closed, under which hypothesis is V_0^X closed as well (as in the case V = Set)?

Second question: supposing that V_0^X is  indeed closed and that V is suitably complete, so that reindexing along f has a right adjoint forall_f, then V_0^X is enriched over V by forall_X(A->B). How is this enrichment related to the usual one of [X,v]  when X is a V-category?

Best regards,

Claudio




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Re: question about monoidal categories

[Steve Lack]

Dear Claudio,

On 18/04/2011, at 8:37 PM, claudio pisani wrote:

> Dear categorists,
> 
> suppose V is a monoidal category, with underlying category V_0, and X is an ordinary category. Then (if I am not mistaken) the functor category V_0^X has a monoidal structure, defined point-wise by that  of V and each functor f:X->Y gives a strong monoidal functor.
> 
> First question : supposing V closed, under which hypothesis is V_0^X closed as well (as in the case V = Set)?
> 

This will be true if V is complete (as you suppose below anyway). Writing, for simplicity, in the case where V is symmetric, the internal
hom [f,g] for functors f,g:X->V_0 is given by the formula (which I hope will be legible)

[f,g]x = int_y [X(x,y).f(y),g(y)]

Here X(x,y) is the hom-set in X, and X(x,y).f(y) is the coproduct of X(x,y) copies of f(y), and int_y denotes the end over all object y in X.

This is a special case of Brian Day's convolution structure where the promonoidal structure on X is the ``cartesian'' one, corresponding
to the cartesian closed structure on [X,Set].

> Second question: supposing that V_0^X is  indeed closed and that V is suitably complete, so that reindexing along f has a right adjoint forall_f, then V_0^X is enriched over V by forall_X(A->B). How is this enrichment related to the usual one of [X,v]  when X is a V-category?
> 

They're the same, essentially because limits commute with limits. 

All the best,

Steve.


> Best regards,
> 
> Claudio
> 


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Re: question about monoidal categories

[claudio pisani]

Dear Steve,

your answer has been very useful to me.

Thus, for a closed complete category V, there are two related doctrines:

1) the V-categories [X,V], indexed by V-Cat;

2) the V-categories V_0^X, indexed by Cat. 

The former is "included" in the latter via the forgetful functor 
V-Cat -> Cat, and the latter has more structure since it is also an indexed  monoidal category.

I would like to know if the second doctrine, and its relationships with the  first one, had been considered explicitly somewhere (perhaps in some paper  by Brian Day?).

Thank you again

Claudio

  

--- Mar 19/4/11, Steve Lack <steve.lack@mq.edu.au> ha scritto:

> Da: Steve Lack <steve.lack@mq.edu.au>
> Oggetto: Re: categories: question about monoidal categories
> A: "claudio pisani" <pisclau@yahoo.it>
> Cc: categories@mta.ca
> Data: Martedì 19 Aprile 2011, 03:40
> Dear Claudio,
> 
> On 18/04/2011, at 8:37 PM, claudio pisani wrote:
> 
>> Dear categorists,
>> 
>> suppose V is a monoidal category, with underlying
> category V_0, and X is an ordinary category. Then (if I am
> not mistaken) the functor category V_0^X has a monoidal
> structure, defined point-wise by that  of V and each
> functor f:X->Y gives a strong monoidal functor.
>> 
>> First question : supposing V closed, under which
> hypothesis is V_0^X closed as well (as in the case V =
> Set)?
>> 
> 
> This will be true if V is complete (as you suppose below
> anyway). Writing, for simplicity, in the case where V is
> symmetric, the internal
> hom [f,g] for functors f,g:X->V_0 is given by the
> formula (which I hope will be legible)
> 
> [f,g]x = int_y [X(x,y).f(y),g(y)]
> 
> Here X(x,y) is the hom-set in X, and X(x,y).f(y) is the
> coproduct of X(x,y) copies of f(y), and int_y denotes the
> end over all object y in X.
> 
> This is a special case of Brian Day's convolution structure
> where the promonoidal structure on X is the ``cartesian''
> one, corresponding
> to the cartesian closed structure on [X,Set].
> 
>> Second question: supposing that V_0^X is  indeed
> closed and that V is suitably complete, so that reindexing
> along f has a right adjoint forall_f, then V_0^X is enriched
> over V by forall_X(A->B). How is this enrichment related
> to the usual one of [X,v]  when X is a V-category?
>> 
> 
> They're the same, essentially because limits commute with
> limits. 
> 
> All the best,
> 
> Steve.
> 
> 
>> Best regards,
>> 
>> Claudio
>> 


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Re: question about monoidal categories

[Richard Garner]

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
>
>
>

Re: question about monoidal categories

[Sam Staton]

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
>
>

Re: question about monoidal categories

[Marco Grandis]

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

Re: question about monoidal categories

[Jeff Egger]

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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Re: question about monoidal categories

[Richard Garner]

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
>
>
>

question about monoidal categories

[Paul B Levy]

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