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, $\mathtt{P}$ and $\mathtt{Q}$, in some monoidal category. Suppose that you’re also given a morphism of operads $\mathtt{Com} \to \mathtt{P}$, where $\mathtt{Com}$ is the operad of commutative algebras. Then Voronov builds a new, bicolored operad $\mathtt{P} \otimes \mathtt{Q}$.

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

$$(\mathtt{P} \otimes \mathtt{Q})(\mathfrak{c}, \dots, \mathfrak{c}; \mathfrak{c}) = \mathtt{P}(n),$$

whereas if any of the colors $x_i$ is not $\mathfrak{c}$,

$$(\mathtt{P} \otimes \mathtt{Q})(x_1, \dots, x_n; \mathfrak{c}) = \varnothing.$$

This is an example of a relative operad (over $\mathtt{P}$), 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 $\mathtt{P}$.

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

$$(\mathtt{P} \otimes \mathtt{Q})(n,m) = \mathtt{P}(m) \otimes \mathtt{Q}(n).$$

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

$$\circ_{i}^{\mathfrak c} : \bigl( \mathtt{P}(m) \otimes \mathtt{Q}(n) \bigr) \otimes \mathtt{P}(m') \to \mathtt{P}(m+m'-1) \otimes \mathtt{Q}(n).$$

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

$$\circ_{i}^{\mathfrak c} : \bigl( \mathtt{P}(m) \otimes \mathtt{Q}(n) \bigr) \otimes \bigl( \mathtt{P}(m') \otimes \mathtt{Q}(n') \bigr) \to \mathtt{P}(m+m') \otimes \mathtt{Q}(n+n'-1).$$

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

$$\begin{align*} \mathtt{P}(m) \otimes \mathtt{P}(m') & \to \mathtt{P}(m+m') \\ p \otimes p' & \mapsto \mu(p, p'). \end{align*}$$

Algebras over Voronov products

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

$$\nu : A \otimes B \to B,$$

corresponding to $\operatorname{id} \otimes \operatorname{id} \in \mathtt{P}(1) \otimes \mathtt{Q}(1)$. This action has to satisfy the following condition, for all $q \in \mathtt{Q}(n)$:

$$q(a_1 \cdot b_1, \dots, a_n \cdot b_n) = (a_1 \dots a_n) \cdot q(b_1, \dots, b_n).$$

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 $\mathtt{SC}$. 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 $\mathtt{Ger}$, the operad encoding Gerstenhaber algebras, and the homology of the little intervals operad is $\mathtt{Ass}$, 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 $\mathtt{Ger} \otimes \mathtt{Ass}$! 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:

$$(a \cdot b) \cdot (a' \cdot b') = aa' \cdot bb', \; \forall a,a' \in A, b,b' \in B.$$

(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 $f(b) = b \cdot 1_A$ (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 $\mathtt{Ger}$ and $\mathtt{Ass}$, consider instead $\mathtt{Ger}_+$ and $\mathtt{Ass}_+$, the operads encoding unital Gerstenhaber algebras and unital associative algebras. There is still a morphism $\mathtt{Com} \to \mathtt{Ger}_+$, so we can build the Voronov product $\mathtt{Ger}_+ \otimes \mathtt{Ass}_+$.

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 $\mathtt{Ger}_+ \otimes_0 \mathtt{Ass}_+$ 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 $A \to Z(B)$ and an action $A \otimes B \to B$, related by:

$$a \cdot b = f(a) \cdot b$$

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 ;).