Concrete category
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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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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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An equivalence between enriched ∞-categories and ∞-categories with weak action Open
We show that an oo-category M with a closed left action of a monoidal oo-category V is completely determined by the V -valued graph of morphism objects resulting from closedness of the action equipped with the structure of a V -enrichment …
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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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Fibration categories are fibrant relative categories Open
A relative category is a category with a chosen class of weak equivalences.\nBarwick and Kan produced a model structure on the category of all relative\ncategories, which is Quillen equivalent to the Joyal model structure on\nsimplicial se…
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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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Abelian right perpendicular subcategories in module categories Open
We show that an abelian category can be exactly, fully faithfully embedded into a module category as the right perpendicular subcategory to a set of modules or module morphisms if and only if it is a locally presentable abelian category wi…
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Homological algebra in characteristic one Open
This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian…
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A categorical approach to abstract convex spaces and interval spaces Open
In this paper, we establish the axiomatic conditions of hull operators and introduce the category of interval spaces. We also investigate their relations with convex spaces from a categorical sense. It is shown that the category CS of conv…
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Mapping cones in the bounded derived category of a gentle algebra Open
In this article we describe the triangulated structure of the bounded derived category of a gentle algebra by describing the triangles induced by the morphisms between indecomposable objects in a basis of their Hom-space.
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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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Lenses and Learners Open
Lenses are a well-established structure for modelling bidirectional transformations, such as the interactions between a database and a view of it. Lenses may be symmetric or asymmetric, and may be composed, forming the morphisms of a monoi…
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Categories of partial equivalence relations as localizations Open
We construct a category of fibrant objects C〈P〉 in the sense of K. Brown from any indexed frame (a kind of indexed poset generalizing triposes) P, and show that its homotopy category is the Barr-exact category C[P] of partial equivalence r…
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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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Model Structures for Correspondences and Bifibrations Open
We study the notion of a bifibration in simplicial sets which generalizes the classical notion of two-sided discrete fibration studied in category theory. If $A$ and $B$ are simplicial sets we equip the category of simplicial sets over $A\…
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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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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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Topology from enrichment: the curious case of partial metrics Open
For any small quantaloid $\Q$, there is a new quantaloid $\D(\Q)$ of diagonals in $\Q$. If $\Q$ is divisible then so is $\D(\Q)$ (and vice versa), and then it is particularly interesting to compare categories enriched in $\Q$ with categori…
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A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS Open
Fried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by show…
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Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans Open
We show that the conditions in Steimle's 'additivity theorem for cobordism categories' can be weakened to only require \emph{locally} (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we comput…
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Goodwillie calculus and Mackey functors Open
We show that the category of $n$-excisive functors from the $\infty$-category of spectra to a target stable $\infty$-category $\mathbf{E}$ is equivalent to the category of $\mathbf{E}$-valued Mackey functors on an indexing category built f…
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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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Homotopy theory of stratified spaces Open
In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those…
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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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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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On the faithfulness of 1-dimensional topological quantum field theories Open
This paper explores 1-dimensional topological quantum field theories.We separately deal with strict and strong 1-dimensional topological quantum field theories.The strict one is regarded as a symmetric monoidal functor between the category…
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The six operations in topology Open
In this paper we show that the six functor formalism for sheaves on locally compact Hausdorff topological spaces, as developed for example in Kashiwara and Schapira's book Sheaves on Manifolds, can be extended to sheaves with values in any…
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The groupoid of finite sets is biinitial in the 2-category of rig\n categories Open
The groupoid of finite sets has a "canonical" structure of a symmetric 2-rig\nwith the sum and product respectively given by the coproduct and product of\nsets. This 2-rig $\\widehat{\\mathbb{F}\\mathbb{S} et}$ is just one of the many\nnon…