The category ${\Theta }_2$, derived modifications, and deformation theory of monoidal categories

Abstract

A complex $C^{\bullet }(C,D)(F,G)(\eta ,\theta )$, generalising the Davydov–Yetter complex of a monoidal category (Davydov (1998) and Yetter (1998)), is constructed. Here, $C,D$ are $\mathbb{k}$-linear (corresp., dg) bicategories, $F,G:C\to D$ are $\mathbb{k}$-linear (corresp., dg) strong functors, and $\eta ,\theta :F\Rightarrow G$ are strong natural transformations. Morally, it is a complex of “derived modifications” $\eta \Rrightarrow \theta$; likewise for the case of dg categories, one has the complex of “derived natural transformations” $F\Rightarrow G$, given by the Hochschild cochain complex of $C$ with coefficients in $C$-bimodule $D(F-,G=)$.
The complex $C^{\bullet }(C,D)(F,G)(\eta ,\theta )$ naturally arises from a 2-cocellular dg vector space $A(C,D)(F,G)(\eta ,\theta ):{\Theta }_2\to C^{\bullet }(\mathbb{k})$, as its ${\Theta }_2$-totalisation (here, ${\Theta }_2$ is the category dual to the category of Joyal 2-disks (Joyal (1997))).
It is shown that for a $\mathbb{k}$-linear monoidal category $C$, the third cohomology vector space $H^3(C^{\bullet }(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id}))$ is isomorphic to the vector space of the outer (modulo twists) infinitesimal deformations of the $\mathbb{k}$-linear monoidal category which we call the full deformations. It means that the following data is to be deformed: (a) the underlying dg category structure, (b) the monoidal product on morphisms (the monoidal product on objects is a set-theoretical datum and is maintained under the deformation), and (c) the associator. The data (a), (b), (c) is subject to the (infinitesimal versions of) numerous monoidal compatibilities, which we interpret as the closeness of the corresponding degree 3 element. Similarly, $H^2(C^{\bullet }(C,D)(F,F)(\mathrm{id},\mathrm{id}))$ is isomorphic to the vector space of the outer infinitesimal deformations of the strong monoidal functor $F$.
A relative totalisation $R{p}_{\ast }A(C,D)(F,F)(\mathrm{id},\mathrm{id})$ along the projection $p:{\Theta }_2\to \Delta$ is defined, and it is shown to be a cosimplicial monoid, which fulfils the Batanin–Davydov 1-commutativity condition (Batanin and Davydov (2023)). Then it follows from loc. cit. that $C^{\bullet }(C,D)(F,F)(\mathrm{id},\mathrm{id})$ is a $C_{\bullet }(E_2;\mathbb{k})$-algebra. Conjecturally, $C^{\bullet }(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id})$ is a $C_{\bullet }(E_3;\mathbb{k})$-algebra; however, the proof requires more sophisticated methods.