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In this article we develop the cotangent complex and (co)homology theories for spectral categories. Along the way, we reproduce standard model structures on spectral categories. As applications, we show that the invariants to descend to stable $infty$-categories and we prove a stabilization result for spectral categories.
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
In this paper, we generalize the combinatorial Laplace operator of Horak and Jost by introducing the $phi$-weighted coboundary operator induced by a weight function $phi$. Our weight function $phi$ is a generalization of Dawsons weighted boundary map
We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these
In the world of chain complexes E_n-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic E_n-homology of an E_n-algebra computes the homology of an n-fold algebraic delooping. The a
We give an explicit point-set construction of the Dennis trace map from the $K$-theory of endomorphisms $Kmathrm{End}(mathcal{C})$ to topological Hochschild homology $mathrm{THH}(mathcal{C})$ for any spectral Waldhausen category $mathcal{C}$. We desc