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Claudio Pisani (Aug 18 2022 at 10:43):Hello everyone,
I just posted the new preprint Operads as double functors,
which may be partly considered as an indexed (instead of fibered) version of the previous Fibered multicategory theory.
The main achievement is the characterization of colored operads (that is, symmetric multicategories)
as product-preserving double (lax) functors from the opposite of the double category of pullback squares in finite sets
to the double category of mappings and spans in sets;
or, equivalently, as normal product-preserving double (lax) functors from the same double category
to that of functors and profunctors.
This may be seen as a generalization of the characterization of commutative monoids
as product-preserving double functors from the opposite of the double category of pullback squares in finite sets
to the double category of squares in sets.
This view suggests the concept of generalized operad (and of generalized commutative monoid)
obtained by replacing, in the above, the category of finite sets with any category C.
For instance, if A is a category with small products (or sums) one gets a generalized operad with C = Set,
and by taking its isomorphism classes one gets the corresponding generalized monoid.
I'm looking forward to receiving your feedback!
Mike Shulman (Aug 18 2022 at 17:55):Does this perspective shed any light on two-sided multicategory-like structures such as polycategories or props?
Claudio Pisani (Aug 19 2022 at 06:31):I'm not sure, but I think that product-preserving double (lax) functors from the opposite of the double category Ref_f
(of mappings and relations on finite sets) to Span(Set), should indeed capture some natural kind of two-sided multicategory-like structure.