2016-02-11: Fredrik Nordvall Forsberg (MSP)

Sat, 21 Oct 2023 21:24:08 +0100

Fredrik Nordvall Forsberg (MSP)

Introduction to (infinity, 1)-categories

Abstract

Infinity-categories simultaneously generalise topological spaces and categories. Intuitively, a (weak) infinity-category should have objects, morphisms, 2-morphisms, 3-morphisms, ... and identity morphisms and composition which is suitably unital and associative up to a higher (invertible) morphism (the number 1 in (infinity, 1)-category means that k-morphisms for k > 1 are invertible) . The trouble begins when one naively tries to make these coherence conditions precise; already 4-categories famously requires 51 pages to define explicitly. Instead, one typically turns to certain "models" of infinity-categories that encode all this data implicitly, usually as some kind of simplicial object with additional properties. I will introduce two such models: quasicategories and complete Segal spaces. If time allows, I will also discuss hopes and dreams about internalising these notions in Type Theory, which should give a satisfactory treatment of category theory in Type Theory without assuming Uniqueness of Identity Proofs.

2016-02-11 11:00:00 UTC
LT1310