Adjoining colimits

TL;DR

This paper develops a higher category theory framework for adjoining colimits via sketches, formalising how to enforce and preserve specified colimits in categories, with applications to presheaves and cocompletion.

Contribution

It introduces a theory of colimit sketches with constructions in higher categories, formalising the process of adjoining and enforcing colimits, and studies their properties and idempotency.

Findings

Established properties of diagrams in infinity-categories as models for presheaves.

Explored conditions for when constructible cocompletion is idempotent.

Introduced categories of presheaves constructible in one step by diagrams.

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.