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The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $operatorname{mod_{mathsf{adm}}}(mathcal{E})$ of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category $mathcal{E}$. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of $mathcal{E}$ are reflected in $operatorname{mod_{mathsf{adm}}}(mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $operatorname{mod_{mathsf{adm}}}(mathcal{E})$ as a subcategory of $operatorname{mod}(mathcal{E})$ when $mathcal{E}$ is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use $operatorname{mod_{mathsf{adm}}}(mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $mathcal{C}$ and certain resolving subcategories of $operatorname{mod}(mathcal{C})$.
Let $mathcal{M}$ be a small $n$-abelian category. We show that the category of finitely presented functors $mod$-$mathcal{M}$ modulo the subcategory of effaceable functors $mod_0$-$mathcal{M}$ has an $n$-cluster tilting subcategory which is equivalen
Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $n$-exact categories and $(n
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 $mathscr{A}$ nat
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine
For a split reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $