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Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,infty)$-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories, which we call magnitude homology (in fact, it is a special sort of Hochschild homology), whose graded Euler characteristic is the magnitude. Magnitude homology of metric spaces generalizes the Hepworth--Willerton magnitude homology of graphs, and detects geometric information such as convexity.
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Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order complexes of posets. In a metric space, an interval (the set of points between two chosen points) has a natural poset structure, which is called the interval poset. Under additional assumptions on sizes of $4$-cuts, we show that the magnitude chain complex can be constructed using tensor products, direct sums and degree shifts from order complexes of interval posets. We give several applications. First, we show the vanishing of higher magnitude homology groups for convex subsets of the Euclidean space. Second, magnitude homology groups carry the information about the diameter of a hole. Third, we construct a finite graph whose $3$rd magnitude homology group has torsion.
We prove a rectification theorem for enriched infinity-categories: If V is a nice monoidal model category, we show that the homotopy theory of infinity-categories enriched in V is equivalent to the familiar homotopy theory of categories strictly enriched in V. It follows, for example, that infinity-categories enriched in spectra or chain complexes are equivalent to spectral categories and dg-categories. A similar method gives a comparison result for enriched Segal categories, which implies that the homotopy theories of n-categories and (infinity,n)-categories defined by iterated infinity-categorical enrichment are equivalent to those of more familia
The invertibility hypothesis for a monoidal model category S asks that localizing an S-enriched category with respect to an equivalence results in an weakly equivalent enriched category. This is the most technical among the axioms for S to be an excellent model category in the sense of Lurie, who showed that the category of S-enriched categories then has a model structure with characterizable fibrant objects. We use a universal property of cubical sets, as a monoidal model category, to show that the invertibility hypothesis is consequence of the other axioms.
Magnitude homology was introduced by Hepworth and Willerton in the case of graphs, and was later extended by Leinster and Shulman to metric spaces and enriched categories. Here we introduce the dual theory, magnitude cohomology, which we equip with the structure of an associative unital graded ring. Our first main result is a recovery theorem showing that the magnitude cohomology ring of a finite metric space completely determines the space itself. The magnitude cohomology ring is non-commutative in general, for example when applied to finite metric spaces, but in some settings it is commutative, for example when applied to ordinary categories. Our second main result explains this situation by proving that the magnitude cohomology ring of an enriched category is graded-commutative whenever the enriching category is cartesian. We end the paper by giving complete computations of magnitude cohomology rings for several large classes of graphs.
In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.
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