The category $\Theta_2$, derived modifications, and deformation theory of monoidal categories

A complex $C^\bullet(C,D)(F,G)(\eta, \theta)$, generalising the Davydov-Yetter complex of a monoidal category, is constructed. Here $C,D$ are $\Bbbk$-linear (dg) monoidal categories, $F,G\colon C\to D$ are $\Bbbk$-linear (dg) strict monoidal functors, $\eta,\theta\colon F\Rightarrow G$ are monoidal 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 $D(F-,G=)$. As well, an intrinsic homological algebra interpretation of $C^\bullet(C,D)(F,G)(\eta,\theta)$ as $RHom$ in an abelian category of 2-bimodules over $C$, is provided. The complex $C^\bullet(C,D)(F,G)(\eta,\theta)$ naturally arises from a 2-cocellular dg vector space $A(C,D)(F,G)(\eta,\theta)\colon \Theta_2\to C^\bullet(\Bbbk)$, as its $\Theta_2$-totalization (here $\Theta_2$ is the category dual to the category of Joyal 2-disks). It is shown that $H^3(C^\bullet(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id})))$ is isomorphic to the vector space of the outer infinitesimal deformations of the $\Bbbk$-linear monoidal category which we call {\it 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), (c) the associator. It is shown that $C^\bullet(C,D)(F,F)(\mathrm{id},\mathrm{id})$ is a homotopy $e_2$-algebra. Conjecturally, $C^\bullet(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id})$ is a homotopy $e_3$-algebra; however the proof requires more sophisticated methods and we hope to complete it in our next paper.