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Register a new userOn the tensor product of linear sites and Grothendieck categories
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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.
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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 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, fails 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].
Let $k$ be a field. We show that locally presentable, $k$-linear categories $mathcal{C}$ dualizable in the sense that the identity functor can be recovered as $coprod_i x_iotimes f_i$ for objects $x_iin mathcal{C}$ and left adjoints $f_i$ from $mathcal{C}$ to $mathrm{Vect}_k$ are products of copies of $mathrm{Vect}_k$. This partially confirms a conjecture by Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also characterize the Grothendieck categories containing an object $x$ with the property that every object is a copower of $x$: they are precisely the categories of non-singular injective right modules over simple, regular, right self-injective rings of type I or III.
Restriction categories were established to handle maps that are partially defined with respect to composition. Tensor topology realises that monoidal categories have an intrinsic notion of space, and deals with objects and maps that are partially defined with respect to this spatial structure. We introduce a construction that turns a firm monoidal category into a restriction category and axiomatise the monoidal restriction categories that arise this way, called tensor-restriction categories.
We introduce a notion of $n$-commutativity ($0le nle infty$) for cosimplicial monoids in a symmetric monoidal category ${bf V}$, where $n=0$ corresponds to just cosimplicial monoids in ${bf V,}$ while $n=infty$ corresponds to commutative cosimplicial monoids. If ${bf V}$ has a monoidal model structure we show (under some mild technical conditions) that the total object of an $n$-cosimplicial monoid has a natural $E_{n+1}$-algebra structure. Our main applications are to the deformation theory of tensor categories and tensor functors. We show that the deformation complex of a tensor functor is a total complex of a $1$-commutative cosimplicial monoid and, hence, has an $E_2$-algebra structure similar to the $E_2$-structure on Hochschild complex of an associative algebra provided by Delignes conjecture. We further demonstrate that the deformation complex of a tensor category is the total complex of a $2$-commutative cosimplicial monoid and, therefore, is naturally an $E_3$-algebra. We make these structures very explicit through a language of Delannoy paths and their noncommutative liftings. We investigate how these structures manifest themselves in concrete examples.
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