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Completing perfect complexes Open
This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented m…
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Parametrized higher category theory Open
We develop foundations for the category theory of $\infty$-categories parametrized by a base $\infty$-category. Our main contribution is a theory of indexed homotopy limits and colimits, which specializes to a theory of $G$-colimits for $G…
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The formal theory of Tannaka duality Open
A Tannakian category is an abelian tensor category equipped with a fiber functor and additional structures which ensure that it is equivalent to the category of representations of some affine groupoid scheme acting on the spectrum of a fie…
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Representations of the oriented skein category Open
The oriented skein category $OS(z,t)$ is a ribbon category which underpins the definition of the HOMFLY-PT invariant of an oriented link, in the same way that the Temperley-Lieb category underpins the Jones polynomial. In this article, we …
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On locally coherent hearts Open
We show that, under particular conditions, if a t-structure in the unbounded\nderived category of a locally coherent Grothendieck category restricts to the\nbounded derived category of its category of finitely presented objects, then\nits …
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A 3-categorical perspective on G-crossed braided categories Open
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…
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Parametrized higher category theory and higher algebra: Exposé I -- Elements of parametrized higher category theory Open
We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we …
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Regular patterns, substitudes, Feynman categories and operads Open
We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the…
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External triangulation of the homotopy category of exact quasi-category Open
Extriangulated categories axiomatize extension-closed subcategories of triangulated categories. We show that the homotopy category of an exact quasi-category can be equipped with a natural extriangulated structure.
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Uniqueness of dg enhancements for the derived category of a Grothendieck category Open
We prove that the derived category of a Grothendieck abelian category has a unique dg enhancement. Under some additional assumptions, we show that the same result holds true for its subcategory of compact objects. As a consequence, we dedu…
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Yoneda lemma for enriched infinity categories Open
We continue the study of enriched infinity categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched infinity categories are associative monoids in an especially designed monoidal category of enrich…
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Open Systems: A Double Categorical Perspective Open
Fong developed `decorated cospans' to model various kinds of open systems: that is, systems with inputs and outputs. In this framework, open systems are seen as the morphisms of a category and can be composed as such, allowing larger open …
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Completing perfect complexes Open
This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented m…
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On equivariant derived categories Open
We study the equivariant category associated to a finite group action on the derived category of coherent sheaves of a smooth projective variety. In particular, we discuss decompositions of the equivariant category, prove the existence of …
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The sl n foam 2-category: A combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality Open
We give an elementary construction of colored sln link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between sln webs; applying a (TQFT-like) representable functor recovers (…
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On the category of finitely generated free groups Open
It is well known that the opposite F^{op} of the category F of finitely generated free groups is a Lawvere theory for groups, and also that F is a free symmetric monoidal category on a commutative Hopf monoid, or, in other words, a PROP fo…
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$t$-Structures with Grothendieck hearts via functor categories Open
We study when the heart of a t-structure in a triangulated category $\mathcal{D}$ with coproducts is AB5 or a Grothendieck category. If $\mathcal{D}$ satisfies Brown representability, a t-structure has an AB5 heart with an injective cogene…
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A homotopy theory of additive categories with suspensions Open
We develop a homotopy theory for additive categories endowed with endofunctors, analogous to the concept of a model structure. We use it to construct the homotopy theory of a Hovey triple (which consists of two compatible complete cotorsio…
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On the Category of EQ-algebras Open
In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculated coproduct and pushout in a …
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Homotopy theory of Moore flows (I) Open
Erratum, 11 July 2022: This is an updated version of the original paper \cite{Moore1} in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix.…
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Hearts of twin Cotorsion pairs on extriangulated categories Open
In this article, we study the heart of a cotorsion pairs on an exact category and a triangulated category in a unified meathod, by means of the notion of an extriangulated category. We prove that the heart is abelian, and construct a cohom…
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Cartesian Integral Categories and Contextual Integral Categories Open
The notion of a Cartesian integral category is introduced and motivated. Morally, this notion should be the coKleisli category of a (tensor) integral category. However, unfortunately, integral categories, as introduced, do not in general h…
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A construction of certain weak colimits and an exactness property of the 2-category of categories Open
Given a 2-category $\mathcal{A}$, a $2$-functor $\mathcal{A} \overset {F} {\longrightarrow} \mathcal{C}at$ and a distinguished 1-subcategory $Σ\subset \mathcal{A}$ containing all the objects, a $σ$-cone for $F$ (with respect to $Σ$) is a l…
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The naive approach for constructing the derived category of a $d$-abelian category fails Open
Let $k$ be a field. In this short note we give an example of a $2$-abelian $k$-category, realized as a $2$-cluster-tilting subcategory of the category $\operatorname{mod}\,A$ of finite dimensional (right) $A$-modules over a finite dimensio…
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Normality and quotient in the category of crossed modules within the category of groups with operations Open
In this paper we define the notions of normal subcrossed module and quotient crossed module within groups with operations; and then give same properties of such crossed modules in groups with operations.
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About the equivalence between monads and monadic functors Open
Given a $\infty$-category X we exhibit the $\infty$-category of right adjoint functors with target X as a localization of the opposite of the $\infty$-category of monads on X. This localization restricts to an equivalence between the $\inf…
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On the category of stratifolds Open
Stratifolds are considered from a categorical point of view. We show among others that the category of stratifolds fully faithfully embeds into the category of ${\mathbb R}$-algebras as does the category of smooth manifolds. We prove that …
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Stacks and sheaves of categories as fibrant objects, II Open
We show that the category of categories bred over a site is a generalized Quillen model category in which the weak equivalences are the local equivalences and the brant objects are the stacks, as they were dened by J. Giraud. The generaliz…
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On the Category of Weakly U-Complexes Open
Motivated by a study of Davvaz and Shabbani which introduced the concept of U-complexes and proposed a generalization on some results in homological algebra, we study thecategory of U-complexes and the homotopy category of U-complexes. In …
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Enriched model categories in equivariant contexts Open
We give a general framework of equivariant model category theory. Our groups G, called Hopf groups, are suitably defined group objects in any well-behaved symmetric monoidal category V. For any V, a discrete group G gives a Hopf group, den…