- I am reading Kontsevich' and Soibelman's paper here. The first chapter is about operads and seems non-standard.
I know the following:
- Let $\mathcal{C}$ be a symmetric monoidal category which is complete and cocomplete and such that the monoidal product is cocontinuous in both arguments. Let $\mathbb{P}$ be the permutation category. Then the presheaf category $[\mathbb{P}^{op},\mathcal{C}]$ (i.e. the catgeory of $\mathcal{C}$-valued species) can be endowed with a monoidal product called the substitution product $\circ$ (see Trimble's notes here, for example.) This substitution product looks explicitly like this $$(F\circ G)[m]:= \coprod_{n\geq 0} F[n]\otimes_{S_n} G^{\otimes_{Day}n}[m].$$
- A symmetric, 1-coloured operad is precisely a monoid in this presheaf category with respect to the substitution product.
- Any operad $P$ in $\mathcal{C}$ gives rise to the action monad $P\circ -$ on $[\mathbb{P}^{op},\mathcal{C}]$. By restricting along the embedding $\mathcal{C}\hookrightarrow [\mathbb{P}^{op},\mathcal{C}]$ (i.e. the diagonal functor) we get a monad on $\mathcal{C}$. Using the above definition of the substitution product this monad looks as follows: $$F[X]:= \coprod_{n\geq 0} F[n]\otimes_{S_n} X^{\otimes n}.$$ (Side remark: I am not sure how the multiplication and the unit of this monad look like.) It is sometimes called an analytic functor/Schur functor/ power series functor. I will use the term analytic functor.
- Back to Kontsevich' and Soibelman's paper:
They additionally assume that $\mathcal{C}$ is closed monoidal, abelian and $k$-linear for a field $k$ of characteristic zero. Then they straight away, without all the motivation I have given above, define analytic functors (see Definition 1 page 4), calling them polynomial functors. Then they claim, without giving details, that the composition of analytic functors is naturally isomorphic to an analytic functor (I suppose this is like the composition of power series?) and that the identity functor is an analytic functor (that is easy to see).
This way the full subcategory of analytic functors becomes a monoidal subcategory of the category of endofunctors on $\mathcal{C}$ with respect to composition. Lets only consider those natural transformations between analytic functors that are induced from natural transformations between the underlying symmetric sequences. As the authors do, let's call this subcategory $\mathcal{PF}$.
Kontsevich and Soibelman then define operads as monoids in the monodial category $\mathcal{PF}$. In other words, to them an operad in $\mathcal{C}$ is simply a monad on $\mathcal{C}$ that is also an analytic functor and whose structure morphisms are morphisms between analytic functors.
- Questions
- It seems to me that what they call an operad is, in the first definition, just the monad attached to an operad, correct? Is this identification of operads and their associated monads harmless?
According to this post, for $\mathcal{C}=Set$, the assignment $$\text{symmetric operad} \mapsto \text{associated analytic functor}$$ is functorial, injective on objects and fully faithful.
- What is the benefit of their definition?