Adjoining colimits
Abstract
This paper develops a theory of colimit sketches "with constructions" in higher category theory, formalising the input to the ubiquitous procedure of adjoining specified "constructible" colimits to a category such that specified "relation" colimits are enforced (or preserved). From a more technical standpoint, sketches are a way to describe dense functors using techniques from the homotopy theory of diagrams. We establish basic properties of diagrams in an infinity-category C as a model for presheaves on C and Bousfield localisations thereof, discuss extensions of functors and adjunctions, and equivalences of sets of diagrams. We introduce categories of presheaves which are "constructible in one step" by a set of diagrams and explore, via well-known examples, when constructible cocompletion is idempotent, i.e. when any iterated construction can be completed in one step.
Cite
@article{arxiv.2111.12117,
title = {Adjoining colimits},
author = {Andrew W. Macpherson},
journal= {arXiv preprint arXiv:2111.12117},
year = {2021}
}
Comments
52 pages. KEYWORDS: diagram, "pursuing stacks", "Theorem A", sketch, localisateur fondamental, localization, rectification, dense functor, cocompletion