# The Voronov Product of Operads

My first real post in a while! It turns out that writing an actual paper (cf. previous blog post) takes a lot of time and effort. Who knew?

The Voronov product of operads is an operation introduced by Voronov in his paper *The Swiss-cheese operad* (he just called it “the product”). It combines an operad and a multiplicative operad to yield a new colored operad; the main example I know is the homology of the Swiss-cheese operad. This is a construction that I use in my preprint Swiss-Cheese operad and Drinfeld center, where as far as I know I coined the name “Voronov product” – I haven’t seen this operation at all outside of Voronov’s paper. I wanted to advertise it a bit because I find it quite interesting and I’m eager to see what people can do with it.

## Voronov products

The setting is as follows. Consider two symmetric one-colored operads, $P$ and $Q$, in some monoidal category. Suppose that you’re also given a morphism of operads $Com→P$, where $Com$ is the operad of commutative algebras. Then Voronov builds a new, bicolored operad $P⊗Q$.

This operad has two colors, $c$ and $o$, that you can think of as “open” and “closed” colors. The operations with closed output are exactly given by $P$, that is:

whereas if any of the colors $x_{i}$ is not $c$,

This is an example of a *relative* operad (over $P$), also known as a *Swiss-cheese type* operad. This type of operad can equivalently be seen as an operad in the category of right modules over $P$.

Composition of such operations is given by the composition of $P$. The operations of $P⊗Q$ with $n$ open inputs, $m$ closed inputs, and an open output, are given by:

There are two kinds of composition to define. To insert an operation with closed output in an operation with open output, one must define:

This composition doesn’t touch the $Q(n)$ factor, and uses the composition of $P$ on the rest. To insert an operation with open output, one must also define:

On the $Q$ factors, this is simply given by the composition of $Q$. On the $P$ factors, recall that we are given a morphism of operads $Com→P$; we thus have some multiplication $μ∈P(2)$, and we can use it to define:

## Algebras over Voronov products

Algebras over $P⊗Q$ have a particularly nice description. Such an algebra is a pair $(A,B)$ where $A$ is an algebra over $P$ and $B$ is an algebra over $Q$. Since we are given a fixed morphism $Com→P$, it follows that $A$ is endowed with a commutative algebra. There is finally an action of $A$ on $B$:

corresponding to $id⊗id∈P(1)⊗Q(1)$. This action has to satisfy the following condition, for all $q∈Q(n)$:

## Example: the homology of the Swiss-cheese operad

The main example of a Voronov product I know is the homology of the Swiss-cheese operad $SC$. Morally speaking, the Swiss-cheese operad is a combination of the little disks operad and the little intervals operad. It makes sense that its homology is given by a combination of their respective homologies.

This is indeed the case. The homology of the little disks operad $Ger$, the operad encoding Gerstenhaber algebras, and the homology of the little intervals operad is $Ass$, the operad encoding associative algebras. If we consider Voronov’s original version of the Swiss-cheese operad, which forbids operations with an open output and no closed input, then the homology is given by the Voronov product $Ger⊗Ass$! That’s as good as can be expected. An algebra over this homology is a pair $(A,B)$ where $A$ is a Gerstenhaber algebra and $B$ is an associative algebra which is also a module over the underlying commutative algebra of $A$, satisfying:

(Here we see the Eckmann–Hilton argument appearing in the background…)

If we now allow operations with an open output and only closed inputs, things get a bit more complicated. The description of the homology of this new operad can be found in the paper “Open-closed homotopy algebras and strong homotopy Leibniz pairs through Koszul operad theory” by Hoefel and Livernet. Just like before, an algebra over this operad is given by a pair consisting of a Gerstenhaber algebra $A$ and an associative algebra $B$. Instead of an action of $A$ on $B$, there is a morphism of commutative algebras from $A$ to the *center* of the algebra $B$. If $B$ is a unital algebra, this is exactly the same thing as before, with $f(b)=b⋅1_{A}$ (and the Eckmann–Hilton argument shows that this lands in the center of $A$).

This new operad can almost be described as the Voronov product of two operads. The remark about unital algebras tips us off. Instead of $Ger$ and $Ass$, consider instead $Ger_{+}$ and $Ass_{+}$, the operads encoding *unital* Gerstenhaber algebras and unital associative algebras. There is still a morphism $Com→Ger_{+}$, so we can build the Voronov product $Ger_{+}⊗Ass_{+}$.

This is not quite right: this encodes a pair consisting of a unital Gerstenhaber algebra, a unital associative algebra, and a central morphism from the former to the latter. To recover the homology of the variant of Swiss-cheese, one simply removes the operations with zero inputs, something I denote $Ger_{+}⊗_{0}Ass_{+}$ in my paper (section 4). When we remove these operations we don’t have units anymore in our algebras, but we keep a central morphism $A→Z(B)$ and an action $A⊗B→B$, related by:

where the first dot is the action of $A$ on $B$ and the second one the multiplication in $B$.

The main motivation for my paper was to try and “lift” this splitting of the homology of Swiss-cheese to the topological level. Due to the non-formality of the Swiss-cheese operad (cf. Livernet, *Non-formality of the Swiss-cheese operad*), it is not actually possible to do; nevertheless I think I succeeded in showing that the Swiss-cheese operad splits as a “shuffled” Voronov product, a notion that I’d like to formalize someday – read my paper for more details ;).