Category theory provides structure for the mathematical world and is seen everywhere in
modern mathematics. With this book, the author bridges the gap between pure category …

We show that, for any simplicial space $ X $, the $\infty $-category of culf maps over $ X $ is
equivalent to the $\infty $-category of right fibrations over $\operatorname {sd}(X) $, the …

This paper continues the research of the author on the homology of cubical and semi-
cubical sets with coefficients in systems. The main result is the theorem that the homology of …

The article investigates the question of under what conditions a functor between small
categories preserves cohomology groups when passing to the inverse image. For example …

We show that the category of decomposition spaces and CULF maps is locally a topos.
Precisely, the slice category over any decomposition space $\mathcal {D} $ is a presheaf …

Extending constructions by Gabriel and Zisman, we develop a functorial framework for the
cohomology and homology of simplicial sets with very general coefficient systems given by …

We introduce twisted arrow categories of operads and of algebras over operads. Up to
equivalence of categories, the simplex category $\Delta $, Segal's category $\Gamma …

Franke's reconstruction functor ℛ is known to provide examples of triangulated equivalences
between homotopy categories of stable model categories, which are exotic in the sense that …

The paper is devoted to homology groups of cubical sets with coefficients in contravariant
systems of abelian groups. The study is based on the proof of the assertion that the …

Abstract In 1996, Jens Franke in an unpublished paper states that the homotopy category of
E (1)-local spectra is equivalent as a triangulated category to D 1 (A), the derived category of …