Dear Jeff, thank your for the very exhaustive answer. Indeed, the condition (*) id_A@e_B@e_A@id_B splits m_{A@B}

(modulo the canonical iso A@B-->A@I@I@B)

was exactly what I needed to complete the proof and your counter-example settles the question. Note in particular that in Chris Heunen's statement CMon can be replaced by Mon. On the other hand I wonder if, for V symmetric, the general statement becomes true with CMon in place of Mon: PROPOSITION 0': Let (V,@,I) be a symmetric monoidal category, and suppose that the forgetful functor U : CMon(V,@,I)--->V is split epi in the (1-)category of (mere) categories and functors.  Then I is initial and @ is cocartesian. (I have seen this statement somewhere, but now I don't remember where). Also note that (if I am not wrong) associtivity does not appear in the proof of your proposition 3. So if UMag is the category of unital magmas in (V,@) we could state PROPOSITION 3': Let (V,@,I) be a monoidal category, and suppose that the forgetful functor U : UMag(V,@,I)--->V is split epi in the (1-)category of (mere) categories and functors.  If (*) holds, then I is initial and @ is cocartesian. Finally, let me explain my interest in that sort of proposition. The idea of a cartesian multicategory has appeared in the literature in various guises; the main idea (it seems to me) is that there are contraction and weakening operation on arrows analogous to those that appear when it comes from a cartesian category (that is C(X,Y;Z) = C(XxY,Z) ). In particular, the following condition should be fulfilled: Representable (in the sense of Hermida) cartesian multicategories correspond to categories with finite products. So I had in mind the following definition: A cartesian multicategory is a multicategory C with a cosimplicial structure on diagonal bimodules. That is a cosimplicial object B_n in Bim(C,C) (the endoprofunctors on the underlying category of C) such that B_n(X,Y) = C(X,..(n times)..,X;Y). It seemed a good idea since 1) It is a simple and natural definition. 2) When applied to the "discrete cocone" multicategory of a category C (with C(X,Y;Z) = C(X;Z)xC(Y;Z) ) it reduces to a Mon-enrichment for C. 3) modulo proposition 0, when C is representable it reduces to existence of finite products on C. So I am a little disappointed that proposition 0 is false; but if proposition 0' holds true then 3) would work with suitable changes. So I would also be interested in any comment on the above proposed notion of cartesian multicategory. Best, Claudio --- Ven 15/3/13, Jeff Egger <jeffegger@yahoo.ca> ha scritto:

Da: Jeff Egger <jeffegger@yahoo.ca> Oggetto: Re: categories: Question on (co)monoids A: "claudio pisani" <pisclau@yahoo.it>, "categories@mta.ca" <categories@mta.ca> Data: Venerdì 15 marzo 2013, 15:22 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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