Question on (co)monoids

Question on (co)monoids

[claudio pisani]

Dear categorists,

in several places I have seen variants of the following statement (or its dual):

If C is a symmetrical monoidal category and every object has a natural monoid structure (that is any map is a monoid  morphism) then C is cocartesian monoidal (tensor = sums).

I have not found any proof of the general statement. 
In fact the tensor product of two objects has obvious maps A -> AxB <-B and for any A -> X <- B it's easy to see that there is at least one factorization AxB -> X.
The problem is unicity. If the monoid structure on objects is unique, then that on AxB (and on I) has to be the usual one and unicity follows.

Question: is the statement true in the above general form or one has to assume that the monoid structure on objects is unique? Or that it is commutative?

I would be grateful for any suggestion or reference.

Claudio




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Re: Question on (co)monoids

[Jeff Egger]

Hi Claudio, 

I was thinking yesterday night about the problem you posed, 

> If C is a symmetrical monoidal category and every object has  a natural monoid 
> structure (that is any map is a monoid  morphism) then C is cocartesian monoidal 
> (tensor = sums).


in light  of the three very different answers which arrived to it on 
Wednesday.  I was sufficiently surprised by my own conclusions that 
I feel a need to share them.


The statement, as given, is a little vague; I interpret it as follows.

PROPOSITION 0: Let (V,@,I) be a symmetric  monoidal category, and
suppose that the forgetful functor U : Mon(V,@,I)--->V is split epi in
the (1-)category of (mere) categories and functors.   Then I is initial
and @ is cocartesian.

[A splitting of U maps each object of V to a specific monoid,
A |-> (A,m_A,e_A), and every arrow to itself.  In other words, every
arrow f:A-->B must be a homomorphism with respect to the specific 
structures (A,m_A,e_A) and (B,m_B,e_B).

It turns out, to my surprise, that Proposition 0 is false: I will
momentarily demonstrate this by means of a concrete counter-example;
but first let me address the issue of what is true.

Chris Heunen suggests patching the statement of Proposition 0 as follows.

PROPOSITION 1: Let (V,@,I) be a symmetric monoidal category, and
suppose that the forgetful functor U : Mon(V,@,I)--->V is invertible
in the (1-)category of (mere)  categories and functors.  Then I is
initial and @ is cocartesian.

Proposition 1 is indeed true, but its utility is somewhat suspect.
To show that U is invertible one must show not only that every object
A carries a monoid structure, but that every monoid structure on A
equals the given one.

Mike Shulman suggests instead (what amounts to): 

PROPOSITION 2: Let (V,@,I) be a symmetric monoidal category, and
suppose that the forgetful functor U : Mon(V,@,I)--->V is split epi in
the (1-)category of tensor categories and functors.  Then I is initial
and @ is cocartesian.

Not only is Proposition 2 true, but its truth implies that of
Proposition 1: the hypothesis of Proposition 2 requires that m_I and
e_I be the canonical isos, and that m_{A@B} and e_{A@B} be related to
m_A@m_B and e_A@e_B in the usual way; namely, via the canonical isos
A@A@B@B-->A@B@A@B and I-->I@I, respectively.  This evidently follows
from the hypothesis of Proposition 1.

But it turns out that the hypothesis of Proposition 2 is still much
stronger than is required.  For instance, those parts which refer to
m_I, e_I, and e_{A@B} are entirely superfluous, and the part referring
to m_{A@B} is only used to establish the following property, which
does not even refer to the symmetry of (V,@,I).

(*) id_A@e_B@e_A@id_B splits m_{A@B}
(modulo the canonical iso A@B-->A@I@I@B)

Hence we arrive at the following.

PROPOSITION 3: Let (V,@,I) be a monoidal category, and suppose that
the forgetful functor U : Mon(V,@,I)--->V is split epi in the
(1-)category of (mere) categories and functors.  If (*) holds, then I
is initial and @ is cocartesian.


Now I will not write out a proof of Proposition 3, which is a tedious
exercise known (at least in spirit) to many.  But I will demonstrate
the necessity of (*) by means of the example promised above.

Let V=Set, I=0, and @ be the tensor product defined by
  A@B = A + B + AxB.
This ``unusual'' symmetric monoidal structure on Set was discussed on
the list  a few years ago in a thread initiated by Peter Selinger.

I don't remember whether it was mentioned at the time, but monoids in
(Set,@,I) are the same thing as semigroups in (Set,x)---i.e.,
semigroups in the most ordinary sense of the word.  For if m : AxA-->A
is associative, then so is  [id_A,id_A,m] : A@A-->A.  Moreover, the
unique map 0-->A is indeed a unit for any map A@A-->A of the form
[id_A,id_A,f].  Conversely, every monoid in (Set,@,0) is of this form.

In this manner, we obtain an isomorphism between Mon(Set,@,I) and Sgp
in the 1-category of mere categories,  which, moreover, commutes with
the two ``underlying set'' functors.  But U:Sgp-->Set is a split epi
that is not invertible; for instance, one has the ``left band           
functor'' Set-->Sgp which assigns to each set A, the semigroup (A,p_l)
with p_l(a,b)=a.  (There is, of course, also a ``right band functor''
which also splits U.)

This is the promised counter-example to Proposition 0.  It is not a
counter-example to Proposition 3, however, because the left band
functor violates  (*).  Let f_{A,B} denote the endomorphism of A@B
defined by composing the following three arrows:
  the canonical iso A@B-->A@I@I@B
  id_A@e_B@e_A@id_B
  m_{A@B}
---then f_{A,B} is a non-trivial idempotent on A@B (not the identity,
as demanded by (*)).  Specificially, it maps a pair (a,b) in third
summand of A@B to its first component a in the first summand of A@B.
Note that A+B is the split of this idempotent.
(Obviously, the right band functor also violates (*).)

This situation is typical: in fact, it is easy to show that the maps
f_{A,B}, as defined above, are always idempotents; moreover, the
following generalisation  of Proposition 3 also holds.

PROPOSITION 4: Let (V,@,I) be a monoidal  category, and suppose that
the forgetful functor U : Mon(V,@,I)--->V is split epi in the
(1-)category of (mere) categories and functors.  Then I is initial,
and if each of the idempotents f_{A,B} is split by some object
S_{A,B}, then V has coproducts given by S_{A,B}.

In general, I guess a monoidal category (V,@,I) for which the
forgetful functor U : Mon(V,@,I)--->V is split epi is what a computer
scientist might call a ``model of sum types without beta-reduction''?
I.e., there are maps fst : A-->A@B and snd : B-->A@B and a copairing
operation [,] satisfying fst[a,b]=a, snd[a,b]=b, but not generally
[fst c,snd c]=c.

That is all.

Cheers,
Jeff.



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Re: Question on (co)monoids

[Dusko Pavlovic]

hi,

the reference for the discussed fact is

@article{FoxT:cartesian,
author = {Thomas Fox},
title = {{Coalgebras and cartesian categories}},
journal = {Communications in Algebra},
volume = {4},
year = {1976},
pages = {665--667},
issue = {7},
doi = {10.1080/00927877608822127},
}

sorry about lagging behind. it is interesting how forgetting leads to new discoveries :)

-- dusko



On Mar 12, 2013, at 11:09 PM, Chris Heunen wrote:

> Dear Claudio,
> 
> One place with a spelled out proof that you could refer to is Theorem
> 2.1 of http://dx.doi.org/10.1016/j.entcs.2008.10.012: Let (C,+,0) be a
> symmetric monoidal category. Then (+,0) provides finite coproducts if
> and only if the forgetful functor cMon(C)->C is an isomorphism of
> categories.
> 
> At that point I, like you, couldn't find any references, but I'm sure
> it is a well-known piece of folklore, and would be interested if you
> can trace its earliest appearance in the literature.
> 
> Best wishes,
> Chris
> 

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