Dear Dusko and Jeff, --- Gio 21/3/13, Dusko Pavlovic <dusko@kestrel.edu> ha scritto:
Da: Dusko Pavlovic <dusko@kestrel.edu> Oggetto: Re: categories: Re: Question on (co)monoids A: "Categories list" <categories@mta.ca> Cc: "claudio pisani" <pisclau@yahoo.it>, "Chris Heunen" <heunen@cs.ox.ac.uk> Data: Giovedì 21 marzo 2013, 18:21 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
In fact, quoting Francois Lamarche (TAC, 2007) "let us mention a well-known fact [Fox76] that we will cannibalize for parts in what follows. Proposition. Given a symmetric monoidal category C, the category of cocommutative comonoids and morphisms of comonoids is a symmetric monoidal category itself [...]. Moreover in this category the tensor is actually the categorical product. Conversely, if every object in a monoidal category is equipped with a natural cocommutative comonoid structure (i.e., every map in the category is a morphism of comonoids) then that monoidal structure is the ordinary categorical product and the comultiplication is the ordinary diagonal." The first part is indeed in Fox's paper, but for the proof he refers to Sweedler's "Hopf algebras" and to a paper by Barr. Both of them leave the proof of (particular cases of) it as an exercise... The second part (which is in fact, in Jeff's terms, proposition 0' of my previous post) is not in Fox paper, which contains instead the (dual of the) following Proposition F "C |-> CMon(C) gives the right adjoint to the inclusion of finite coproduct categories into monoidal categories (with strong monoidal functor)" Thus, I suppose Proposition 0' follows easily from Proposition F. Why? --- Mar 19/3/13, Jeff Egger <jeffegger@yahoo.ca> ha scritto:
Da: Jeff Egger <jeffegger@yahoo.ca> Oggetto: categories: Re: Question on (co)monoids (and on cartesian multicategories) A: "claudio pisani" <pisclau@yahoo.it> Data: Martedì 19 marzo 2013, 14:54
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) ).
Yes, I am most familiar with the definition contained on p.33 of
(thanks for this helpful reference)
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).
If I replace the cosimplicial structure (an action of Delta) by an action of FSet (finite sets) the definition becomes very similar to that of FP operad in Miles Gould thesis. The difference is that I require the action only on diagonal arrows. Similarly, if one is interested just in symmetry, one could ask for an action of bijections on diagonal arrows and the corresponding representable concept would be a monoidal category C with a natural involutive isomorphism X@X -> X@X : C -> C ... After all, this is what is needed to talk about commutative monoids in C. Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]