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Register a new userEquivalences between localisations of categories provided by replacements
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Authors
Sebastian Thomas
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We give a characterisation of functors whose induced functor on the level of localisations is an equivalence and where the isomorphism inverse is induced by some kind of replacements such as projective resolutions or cofibrant replacements.
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Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. This work provides a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user friendly resource for beginners and experts alike.
This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via universal properties. In this paper, we introduce enriched presheaves on an enriched infinity category. We prove analogues of most familiar properties of presheaves. For example, we compute limits and colimits of presheaves, prove that all presheaves are colimits of representable presheaves, and prove a version of the Yoneda lemma.
We study convergent (terminating and confluent) presentations of n-categories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for n-categories, generalizing the one introduced by Squier for word rewriting systems. We characterize this property by using the notion of critical branching. In particular, we define sufficient conditions for an n-category to have finite derivation type. Through examples, we present several techniques based on derivations of 2-categories to study convergent presentations by 3-polygraphs.
It is well known that a resolving subcategory $mathcal{A}$ of an abelian subcategory $mathcal{E}$ induces several derived equivalences: a triangle equivalence $mathbf{D}^-(mathcal{A})to mathbf{D}^-(mathcal{E})$ exists in general and furthermore restricts to a triangle equivalence $mathbf{D}^{mathsf{b}}(mathcal{A})to mathbf{D}^{mathsf{b}}(mathcal{E})$ if $operatorname{res.dim}_{mathcal{A}}(E)<infty$ for any object $Ein mathcal{E}$. If the category $mathcal{E}$ is uniformly bounded, i.e. $operatorname{res.dim}_{mathcal{A}}(mathcal{E})<infty$, one obtains a triangle equivalence $mathbf{D}(mathcal{A})to mathbf{D}(mathcal{E})$. In this paper, we show that all of the above statements hold for preresolving subcategories of (one-sided) exact categories. By passing to a one-sided language, one can remove the assumption that $mathcal{A}subseteq mathcal{E}$ is extension-closed completely from the classical setting, yielding easier criteria and more examples. To illustrate this point, we consider the Isbell category $mathcal{I}$ and show that $mathcal{I}subseteq mathsf{Ab}$ is preresolving but $mathcal{I}$ cannot be realized as an extension-closed subcategory of an exact category. We also consider a criterion given by Keller to produce derived equivalences of fully exact subcategories. We show that this criterion fits into the framework of preresolving subcategories by considering the relative weak idempotent completion of said subcategory.
We recall several categories of graphs which are useful for describing homotopy-cohere
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