Univalent higher categories via complete Semi-Segal types

Tamsamani Categories, Segal Categories, and Applications

Two talks by Bertrand Toën at the IMA 2004 Summer programme on n-Categories: foundations and applications. (Notes taken by Joachim Kock, revised by B.T.) This version 2 (June 2005) is identical to the original except that a correction to 1.3 is given in a footnote.

1.2 Segal Spaces and Complete Segal Spaces.............................. 11

1.2 Segal Spaces and Complete Segal Spaces.............................. 11

Univalent categories and the Rezk completion

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of " category " for which equality and equivalence of categories agree. Such categories satisfy a version of the Univalence Axiom, saying that the type of isomorphisms between any two objec...

Almost Complete Cluster Tilting Objects in Generalized Higher Cluster Categories

We study higher cluster tilting objects in generalized higher cluster categories arising from dg algebras of higher Calabi-Yau dimension. Taking advantage of silting mutations of Aihara-Iyama, we obtain a class of m-cluster tilting objects in generalized m-cluster categories. For generalized m-cluster categories arising from strongly (m + 2)-Calabi-Yau dg algebras, by using truncations of minim...