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Register a new userAnalogues of centralizer subalgebras for fiat 2-categories and their 2-representations
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The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive $2$-representations with a fixed apex $mathcal{J}$ of a fiat $2$-category $cC$ and the set of equivalence classes of faithful simple transitive $2$-representations of the fiat $2$-subquotient of $cC$ associated with a diagonal $mathcal{H}$-cell in $mathcal{J}$. As an application, we classify simple transitive $2$-representations of various categories of Soergel bimodules, in particular, completing the classification in types $B_3$ and $B_4$.
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In this article we analyze the structure of $2$-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under which condition the resulting $2$-category is fiat (in the sense of cite{MM1}) and classify simple transitive $2$-representations of this $2$-category (under some mild technical assumption). We also study several classes of examples in detail.
Let $mathscr{F}$ be an $(n+2)$-angulated Krull-Schmidt category and $mathscr{A} subset mathscr{F}$ an $n$-extension closed, additive and full subcategory with $operatorname{Hom}_{mathscr{F}}(Sigma_n mathscr{A}, mathscr{A}) = 0$. Then $mathscr{A}$ naturally carries the structure of an $n$-exact category in the sense of Jasso, arising from short $(n+2)$-angles in $mathscr{F}$ with objects in $mathscr{A}$ and there is a binatural and bilinear isomorphism $operatorname{YExt}^{n}_{(mathscr{A},mathscr{E}_{mathscr{A}})}(A_{n+1},A_0) cong operatorname{Hom}_{mathscr{F}}(A_{n+1}, Sigma_n A_{0})$ for $A_0, A_{n+1} in mathscr{A}$. For $n = 1$ this has been shown by Dyer and we generalize this result to the case $n > 1$. On the journey to this result, we also develop a technique for harvesting information from the higher octahedral axiom (N4*) as defined by Bergh and Thaule. Additionally, we show that the axiom (F3) for pre-$(n+2)$-angulated categories, introduced by Geiss, Keller and Oppermann and stating that a commutative square can be extended to a morphism of $(n+2)$-angles, implies a stronger version of itself.
The notion of right semi-equivalence in a right $(n+2)$-angulated category is defined in this article. Let $mathscr C$ be an $n$-exangulated category and $mathscr X$ is a strongly covariantly finite subcategory of $mathscr C$. We prove that the standard right $(n+2)$-angulated category $mathscr C/mathscr X$ is right semi-equivalence under a natural assumption. As an application, we show that a right $(n+2)$-angulated category has an $n$-exangulated structure if and only if the suspension functor is right semi-equivalence. Besides, we also prove that an $n$-exangulated category $mathscr C$ has the structure of a right $(n+2)$-angulated category with right semi-equivalence if and only if for any object $Xinmathscr C$, the morphism $Xto 0$ is a trivial inflation.
Let $mathscr C$ be a Krull-Schmidt $(n+2)$-angulated category and $mathscr A$ be an $n$-extension closed subcategory of $mathscr C$. Then $mathscr A$ has the structure of an $n$-exangulated category in the sense of Herschend-Liu-Nakaoka. This construction gives $n$-exangulated categories which are not $n$-exact categories in the sense of Jasso nor $(n+2)$-angulated categories in the sense of Geiss-Keller-Oppermann in general. As an application, our result can lead to a recent main result of Klapproth.
Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of type $mathfrak{sl}_2$ at prime roots of unity by Elias-Qi [Advances in Mathematics 288 (2016)]. Passing to stable 2-representations gives a way to construct triangulated 2-representations, but our main focus is on working with p-dg enriched 2-representations that should be seen as a p-dg enhancement of these triangulated ones.
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