Dear All, In the chapter 3 of their book "Monoidal functor, species and Hopf algebras" http://www.math.tamu.edu/~maguiar/ Aguiar and Mahajan introduces 4 kinds of monoidal functors: 1) strong monoidal 2) lax monoidal 3) colax monoidal 4) bilax monoidal A monoid in a monoidal category C is a lax monoidal functor 1-->C, a comonoid is a colax monoidal functor 1-->C and a bimonoid is a bilax monoidal functor 1-->C. I wonder who first introduced the notion of bilax monoidal functor and when? An example of bilax monoidal functor is the singuler chain complex functor from spaces to chain complexes. The bilax structure is provided by the Eilenberg-MacLane map together with the Alexander-Whitney map. Best, AJ -------- Message d'origine-------- De: categories@mta.ca de la part de Steve Lack Date: jeu. 06/05/2010 19:02 À: Fred E.J. Linton; categories Objet : Re: categories: Q. about monoidal functors Dear Fred, Such a T is called a symmetric monoidal functor. Example: let _A_ be Set with the cartesian monoidal structure. Let M be a monoid and let T be the functor Set->Set sending X to MxX (which I'll write as MX). This functor T is monoidal via the map MXMY->MXY sending (m,x,n,y) to (mn,x,y). It is symmetric monoidal iff M is commutative. Steve Lack. On 6/05/10 4:01 PM, "Fred E.J. Linton" <fejlinton@usa.net> wrote:
Suppose _A_ is a symmetric monoidal category in the sense of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_ a monoidal functor.
What, if anything, is known, where τ: X ⊗ Y --> Y ⊗ X is the symmetry structure on the (symmetric) tensor product ⊗, as to whether
[T_X,Y: TX ⊗ TY --> T(X ⊗ Y)] and [T(τ_X,Y): T(X ⊗ Y) --> T(Y ⊗ X)]
have the same composition as have
[τ_TX,TY: TX ⊗ TY --> TY ⊗ TX] and [T_Y,X: TY ⊗ TX --> T(Y ⊗ X)] ?
TIA for any relevant information and/or references thereto.
Cheers, -- Fred
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