Non-permutative operads in a presheaf topos
Mark Weber – 22 September 1999
The higher operads of Michael Batanin can be thought of as a type of universal algebra where the arities are planar trees instead of natural numbers. The goal of these talks was to make this description precise, by giving a general definition of universal algebra and non-permutative operad for a presheaf topos, and indicating how in the case of globular sets, Batanin's operads are special types of non-permutative operads in this sense.
Fix a small category C.
Definition: An arity for the presheaf topos [C^op, Set] is a monoidally dense fully faithful functor I : A ---> [C^op, Set] where A is small.
That is, the essential image of the functor Lan_I : [A, [C^op, Set]] ---> End[C^op, Set] contains the identity and is closed under composition. This functor induces a "substitution" monoidal structure on its domain in such a way that it becomes strong monoidal (wrt composition in the codomain). One obtains endofunctors arising from collections and monads arising from non-permutative operads by considering the connected limit preserving endofunctors in the image of Lan_I.
The classical example is where C =1, A = Set_f and I is the inclusion. The Batanin example will be C = G and A = tr = full subcategory of Glob consisting of those globular sets of the form T* for some tree T (see Batanin: "Monoidal Globular Categories as a Natural Environment for the Theory of Weak n-Categories" for the notation).
In order to see Batanin's collections as collections in the above sense, we obtain a general characterisation of the connected limit preserving functors in the image of Lan_I, where I is as above except that it needn't be monoidally dense.
Preliminary to the next definition, consider the functor [C/-, K] : C^op ---> Cat where K is an arbitrary category. For example, putying C = G one obtains cospan(K) in the sense of Batanin (ie the globular category of higher cospans, and if K has finite colimits then cospan(K) is actually a monoidal globular category).
Definition: The category C-Fam(A) has as objects, arrows f : L ---> [C/-, A] in [C^op, Cat], where L is discrete (ie in [C^op, Set]). An arrow (L,f) ---> (L', f ') consists of an arrow g : L ---> L' and a 2-cell g' : f 'g ==> f in [C^op, Cat].
Example: 1-Fam(Set_f) = Fam(Set_f) in the usual sense.
Let A contain all its retracts (ie a retract of something in A is isomorphic to something in A). Then one obtains:
Theorem: There is an equivalence of categories between C-Fam(A) and the full subcategory of End[C^op, Set] consisting of the (endofunctors arising from) collections.
The case C = 1 of this theorem is contained in Carboni & Johnstone: "Connected Limits, Familial Representability and Artin-Glueing".
As a corollary, a globular collection amounts to a globular functor L ---> cospan(tr) where L is a globular set. One can now recognise Batanin collections as such globular functors whose image consists of cospans that are canonical in a certain sense.