Re: bilax_monoidal_functors?=

[Richard Garner <rhgg2@hermes.cam.ac.uk>]

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Dear André

> In the chapter 4 of the latest version of their book
>
> http://www.math.tamu.edu/~maguiar/a.pdf
>
> Aguiar and Mahajan introduce a notion of P-monoidal functor
> for P is an operad.
>
> ...
>
> The notion of P-monoidal functor for P a PROP
> is not defined in their book.
>
> Any idea?

The notion of P-monoidal functor C --> D, for P an operad, 
may be described most expediently when D has small colimits, 
and these distribute over tensors: for then the functor 
category [C,D] is itself monoidal under Day convolution, and 
a P-monoidal functor is nothing but a P-algebra in this 
functor category.

Such a definition admits an obvious generalisation to the 
case where P is not an operad, but merely a PROP; however, it 
seems to me that such a generalisation has no force. For 
instance, when P is the PROP for coalgebras the notion of 
P-monoidal functor is nothing like that of an oplax monoidal 
functor. The problem is one of variance; in giving a 
comultiplication F -> F * F one is required to map into a 
coend, which is back-to-front.

One way of throwing this into focus is by considering the 
case where D has no colimits to speak of, so that the functor 
category [C,D] is not monoidal, but merely a multicategory. 
In any multicategory, one may still speak of a P-algebra for 
an operad---thereby allowing the notion of P-monoidal 
morphism of Aguiar and Mahajan to find its fully general 
expression---but the notion of P-algebra for an arbitrary 
PROP no longer makes sense.

The moral is that a PROP in general may be built from 
components which originate "in algebra" and components which 
originate "in coalgebra"; or, indeed, from components which 
originate in neither. It is, I think, only when all 
components originate "in algebra"---which is to say that the 
PROP is an operad---that the notion of P-monoidal functor is 
mathematically sensible.

However, all is not lost; for many of the PROPs of interest 
are not just PROPs, but instances of some smaller notion 
which allows "algebraic" and "coalgebraic" components 
interacting according to some particular discipline. For 
example, there is Gan's notion of dioperad, which is 
essentially that of a one-object polycategory. The PROP for a 
Frobenius algebra is an example of such a dioperad. Now, we 
may speak of models for a dioperad in any polycategory; and 
in particular, the functor category [C,D] between two 
monoidal categories bears such a polycategorical structure, 
wherein a model for the dioperad for Frobenius algebras is 
precisely a Frobenius monoidal functor. Jeff Egger has 
studied circumstances under which this polycategorical 
structure of [C,D] is representable, and almost certainly 
discusses this example of Frobenius monoidal functors; but he 
is much more qualified than me to speak on this!

Richard


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