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Register a new userOn the uniqueness of infinity-categorical enhancements of triangulated categories
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Benjamin Antieau
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We study the problem of when triangulated categories admit unique infinity-categorical enhancements. Our results use Luries theory of prestable infinity-categories to give conceptual proofs of, and in many cases strengthen, previous work on the subject by Lunts--Orlov and Canonaco--Stellari. We also give a wide range of examples involving quasi-coherent sheaves, categories of almost modules, and local cohomology to illustrate the theory of prestable infinity-categories. Finally, we propose a theory of stable $n$-categories which would interpolate between triangulated categories and stable infinity-categories.
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The goal of the article is to better understand cosupport in triangulated categories since it is still quite mysterious. We study boundedness of local cohomology and local homology functors using Koszul objects, give some characterizations of cosupport and get some results that, in special cases, recover and generalize the known results about the usual cosupport. Also we include some computations of cosupport, settle the comparison of support and cosupport of cohomologically finite objects. Finally, we assign to any object of the category a subset of $mathrm{Spec}R$, called the big cosupport.
We make precise the analogy between Goodwillies calculus of functors in homotopy theory and the differential calculus of smooth manifolds by introducing a higher-categorical framework of which both theories are examples. That framework is an extension to infinity-categories of the tangent categories of Cockett and Cruttwell (introduced originally by Rosicky). A tangent structure on an infinity-category X consists of an endofunctor on X, which plays the role of the tangent bundle construction, together with various natural transformations that mimic structure possessed by the ordinary tangent bundles of smooth manifolds and that satisfy similar conditions. The tangent bundle functor in Goodwillie calculus is Luries tangent bundle for infinity-categories, introduced to generalize the cotangent complexes of Andre, Quillen and Illusie. We show that Luries construction admits the additional structure maps and satisfies the conditions needed to form a tangent infinity-category, which we refer to as the Goodwillie tangent structure on the infinity-category of infinity-categories. Cockett and Cruttwell (and others) have started to develop various aspects of differential geometry in the abstract context of tangent categories, and we begin to apply those ideas to Goodwillie calculus. For example, we show that the role of Euclidean spaces in the calculus of manifolds is played in Goodwillie calculus by the stable infinity-categories. We also show that Goodwillies n-excisive functors are the direct analogues of n-jets of smooth maps between manifolds; to state that connection precisely, we develop a notion of tangent (infinity, 2)-category and show that Goodwillie calculus is best understood in that context.
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.
In this paper we give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG-enhancement. The only previous examples of triangulated categories without a model have been constructed by Muro, Schwede and Strickland. These examples are however not linear over a field.
Let $mathcal{C}$ be a triangulated category. We first introduce the notion of balanced pairs in $mathcal{C}$, and then establish the bijective correspondence between balanced pairs and proper classes $xi$ with enough $xi$-projectives and enough $xi$-injectives. Assume that $xi:=xi_{mathcal{X}}=xi^{mathcal{Y}}$ is the proper class induced by a balanced pair $(mathcal{X},mathcal{Y})$. We prove that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is an extriangulated category. Moreover, it is proved that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is a triangulated category if and only if $mathcal{X}=mathcal{Y}=0$; and that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is an exact category if and only if $mathcal{X}=mathcal{Y}=mathcal{C}$. As an application, we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.
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