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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]