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We present an algorithm for approximating linear categories of partitions (of sets). We report on concrete computer experiments based on this algorithm which we used to obtain first examples of so-called non-easy linear categories of partitions. All of the examples that we constructed are proven to be indeed new and non-easy. We interpret some of the new categories in terms of quantum group anticommutative twists.
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This work is the first one in a series, in which we develop a mathematical theory of enriched (braided) monoidal categories and their representations. In this work, we introduce the notion of the $E_0$-center ($E_1$-center or $E_2$-center) of an enriched (monoidal or braided monoidal) category, and compute the centers explicitly when the enriched (braided monoidal or monoidal) categories are obtained from the canonical constructions. These centers have important applications in the mathematical theory of gapless boundaries of 2+1D topological orders and that of topological phase transitions in physics. They also play very important roles in the higher representation theory, which is the focus of the second work in the series.
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.
A braided monoidal category may be considered a $3$-category with one object and one $1$-morphism. In this paper, we show that, more generally, $3$-categories with one object and $1$-morphisms given by elements of a group $G$ correspond to $G$-crossed braided categories, certain mathematical structures which have emerged as important invariants of low-dimensional quantum field theories. More precisely, we show that the 4-category of $3$-categories $mathcal{C}$ equipped with a 3-functor $mathrm{B}G to mathcal{C}$ which is essentially surjective on objects and $1$-morphisms is equivalent to the $2$-category of $G$-crossed braided categories. This provides a uniform approach to various constructions of $G$-crossed braided categories.
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].
We study plane partitions satisfying condition $a_{n+1,m+1}=0$ (this condition is called pit) and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane partitions label the basis vectors in certain representations of quantum toroidal $mathfrak{gl}_1$ algebra, therefore our formulas can be interpreted as the characters of these representations. The resulting formulas resemble formulas for characters of tensor representations of Lie superalgebra $mathfrak{gl}_{m|n}$. We discuss representation theoretic interpretation of our formulas using $q$-deformed $W$-algebra $mathfrak{gl}_{m|n}$.
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