In this paper, which is subsequent to our previous paper [PS] (but can be read independently from it), we continue our study of the closed model structure on the category $mathrm{Cat}_{mathrm{dgwu}}(Bbbk)$ of small weakly unital dg categories (in the
sense of Kontsevich-Soibelman [KS]) over a field $Bbbk$. In [PS], we constructed a closed model structure on the category of weakly unital dg categories, imposing a technical condition on the weakly unital dg categories, saying that $mathrm{id}_xcdot mathrm{id}_x=mathrm{id}_x$ for any object $x$. Although this condition led us to a great simplification, it was redundant and had to be dropped. Here we get rid of this condition, and provide a closed model structure in full generality. The new closed model category is as well cofibrantly generated, and it is proven to be Quillen equivalent to the closed model category $mathrm{Cat}_mathrm{dg}(Bbbk)$ of (strictly unital) dg categories over $Bbbk$, given by Tabuada [Tab1]. Dropping the condition $mathrm{id}_x^2=mathrm{id}_x$ makes the construction of the closed model structure more distant from loc.cit., and requires new constructions. One of them is a pre-triangulated hull of a wu dg category, which in turn is shown to be a wu dg category as well. One example of a weakly unital dg category which naturally appears is the bar-cobar resolution of a dg category. We supply this paper with a refinement of the classical bar-cobar resolution of a unital dg category which is strictly unital (appendix B). A similar construction can be applied to constructing a cofibrant resolution in $mathrm{Cat}_mathrm{dgwu}(Bbbk)$.
In our recent papers [Sh1,2], we introduced a {it twisted tensor product} of dg categories, and provided, in terms of it, {it a contractible 2-operad $mathcal{O}$}, acting on the category of small dg categories, in which the natural transformations a
re derived. We made use of some homotopy theory developed in [To] to prove the contractibility of the 2-operad $mathcal{O}$. The contractibility is an important issue, in vein of the theory of Batanin [Ba1,2], according to which an action of a contractible $n$-operad on $C$ makes $C$ a weak $n$-category. In this short note, we provide a new elementary proof of the contractibility of the 2-operad $mathcal{O}$. The proof is based on a direct computation, and is independent from the homotopy theory of dg categories (in particular, it is independent from [To] and from Theorem 2.4 of [Sh1]).
In this paper, we study weakly unital dg categories as they were defined by Kontsevich and Soibelman [KS, Sect.4]. We construct a cofibrantly generated Quillen model structure on the category $mathrm{Cat}_{mathrm{dgwu}}(Bbbk)$ of small weakly unital
dg categories over a field $Bbbk$. Our model structure can be thought of as an extension of the model structure on the category $mathrm{Cat}_{mathrm{dg}}(Bbbk)$ of (strictly unital) small dg categories over $Bbbk$, due to Tabuada [Tab]. More precisely, we show that the imbedding of $mathrm{Cat}_{mathrm{dg}}(Bbbk)$ to $mathrm{Cat}_{mathrm{dgwu}}(Bbbk)$ is a right adjoint of a Quillen pair of functors. We prove that this Quillen pair is, in turn, a Quillen equivalence. In course of the proof, we study a non-symmetric dg operad $mathcal{O}$, governing the weakly unital dg categories, which is encoded in the Kontsevich-Soibelman definition. We prove that this dg operad is quasi-isomorphic to the operad $mathrm{Assoc}_+$ of unital associative algebras.
It is well-known that the pre-2-category $mathscr{C}at_mathrm{dg}^mathrm{coh}(k)$ of small dg categories over a field $k$, with 1-morphisms defined as dg functors, and with 2-morphisms defined as the complexes of coherent natural transformations, fai
ls to be a strict 2-category. In [T2], D.Tamarkin constructed a contractible 2-operad in the sense of M.Batanin [Ba3], acting on $mathscr{C}at_mathrm{dg}^mathrm{coh}(k)$. According to Batanin loc.cit., it is a possible way to define a weak 2-category. In this paper, we provide a construction of {it another} contractible 2-operad $mathcal{O}$, acting on $mathscr{C}at_mathrm{dg}^mathrm{coh}(k)$. Our main tool is the {it twisted tensor product} of small dg categories, introduced in [Sh3]. We establish a one-side associativity for the twisted tensor product, making $(mathscr{C}at_mathrm{dg}^mathrm{coh}(k),overset{sim}{otimes})$ a skew monoidal category in the sense of [LS], and construct a {it twisted composition} $mathscr{C}oh_mathrm{dg}(D,E)overset{sim}{otimes}mathscr{C}oh_mathrm{dg}(C,D)tomathscr{C}oh_mathrm{dg}(C,E)$, and prove some compatibility between these two structures. Taken together, the two structures give rise to a 2-operad $mathcal{O}$, acting on $mathscr{C}at_mathrm{dg}^mathrm{coh}(k)$. Its contractibility is a consequence of a general result of [Sh3].
Given two small dg categories $C,D$, defined over a field, we introduce their (non-symmetric) twisted tensor product $Coverset{sim}{otimes} D$. We show that $-overset{sim}{otimes} D$ is left adjoint to the functor $Coh(D,-)$, where $Coh(D,E)$ is the
dg category of dg functors $Dto E$ and their coherent natural transformations. This adjunction holds in the category of small dg categories (not in the homotopy category of dg categories $mathrm{Hot}$). We show that for $C,D$ cofibrant, the adjunction descends to the corresponding adjunction in the homotopy category. Then comparison with a result of To{e}n shows that, for $C,D$ cofibtant, $Coverset{sim}{otimes} D$ is isomorphic to $Cotimes D$, as an object of the homotopy category $mathrm{Hot}$.
It is well-known that the Kontsevich formality [K97] for Hochschild cochains of the polynomial algebra $A=S(V^*)$ fails if the vector space $V$ is infinite-dimensional. In the present paper, we study the corresponding obstructions. We construct an $L
_infty$ structure on polyvector fields on $V$ having the even degree Taylor components, with the degree 2 component given by the Schouten-Nijenhuis bracket, but having as well higher non-vanishing Taylor components. We prove that this $L_infty$ algebra is quasi-isomorphic to the corresponding Hochschild cochain complex. We prove that our $L_infty$ algebra is $L_infty$ quasi-isomorphic to the Lie algebra of polyvector fields on $V$ with the Schouten-Nijenhuis bracket, if $V$ is finite-dimensional.
In this paper, we prove that there is a canonical homotopy $(n+1)$-algebra structure on the shifted operadic deformation complex $Def(e_ntomathcal{P})[-n]$ for any operad $mathcal{P}$ and a map of operads $fcolon e_ntomathcal{P}$. This result general
izes the result of [T2], where the case $mathcal{P}=mathrm{End}_{Op}(X)$ was considered. Another more computational proof of the same statement was recently sketched in [CW]. Our method combines the one of [T2] with the categorical algebra on the category of symmetric sequences, introduced in [R] and further developed in [KM] and [Fr1]. We define suitable deformation functors on $n$-coalgebras, which are considered as the non-commutative base of deformation, prove their representability, and translate properties of the functors to the corresponding properties of the representing objects. A new point, which makes the method more powerful, is to consider the argument of our deformation theory as an object of the category of symmetric sequences of dg vector spaces, not as just a single dg vector space.
We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally
presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.
In our recent paper [Sh1] a version of the generalized Deligne conjecture for abelian $n$-fold monoidal categories is proven. For $n=1$ this result says that, given an abelian monoidal $k$-linear category $mathscr{A}$ with unit $e$, $k$ a field of ch
aracteristic 0, the dg vector space $mathrm{RHom}_{mathscr{A}}(e,e)$ is the first component of a Leinster 1-monoid in $mathscr{A}lg(k)$ (provided a rather mild condition on the monoidal and the abelian structures in $mathscr{A}$, called homotopy compatibility, is fulfilled). In the present paper, we introduce a new concept of a ${it graded}$ Leinster monoid. We show that the Leinster monoid in $mathscr{A}lg(k)$, constructed by a monoidal $k$-linear abelian category in [Sh1], is graded. We construct a functor, assigning an algebra over the chain operad $C(E_2,k)$, to a graded Leinster 1-monoid in $mathscr{A}lg(k)$, which respects the weak equivalences. Consequently, this paper together with loc.cit. provides a complete proof of the generalized Deligne conjecture for 1-monoidal abelian categories, in the form most accessible for applications to deformation theory (such as Tamarkins proof of the Kontsevich formality).
We provide a more economical refined version of Evrards categorical cocylinder factorization of a functor [Ev1,2]. We show that any functor between small categories can be factored into a homotopy equivalence followed by a (co)fibred functor which sa
tisfies the (dual) assumption of Quillens Theorem B.
