Adjoining colimits

Andrew W. Macpherson

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arXiv · 2111.12117

Adjoining colimits

Andrew W. Macpherson

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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.

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BibTeX RIS

Andrew W. Macpherson. 2021-11-23. Adjoining colimits. https://arxiv.org/abs/2111.12117

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