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 ;).