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A complex $C^\bullet(C,D)(F,G)(\eta, \theta)$, generalising theDavydov-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 monoidalnatural transformations. Morally, it is a complex of ``derived modifications''$\eta \Rrightarrow \theta$, likewise for the case of dg categories one has thecomplex of ``derived natural transformations'' $F\Rightarrow G$, given by theHochschild 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 of2-bimodules over $C$, is provided.  The complex $C^\bullet(C,D)(F,G)(\eta,\theta)$ naturally arises from a2-cocellular dg vector space $A(C,D)(F,G)(\eta,\theta)\colon \Theta_2\toC^\bullet(\Bbbk)$, as its $\Theta_2$-totalization (here $\Theta_2$ is thecategory dual to the category of Joyal 2-disks).  It is shown that$H^3(C^\bullet(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id})))$ isisomorphic to the vector space of the outer infinitesimal deformations of the$\Bbbk$-linear monoidal category which we call {\it full} deformations. Itmeans that the following data is to be deformed: (a) the underlying dg categorystructure, (b) the monoidal product on morphisms (the monoidal product onobjects 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 ahomotopy $e_3$-algebra; however the proof requires more sophisticated methodsand we hope to complete it in our next paper.

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