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Colimits and cocompletions in internal higher category theory

Louis Martini, Sebastian Wolf

Abstract

We develop a number of basic concepts in the theory of categories internal to an $\infty$-topos. We discuss adjunctions, limits and colimits as well as Kan extensions for internal categories, and we use these results to prove the universal property of internal presheaf categories. We furthermore construct the free cocompletion of an internal category by colimits that are indexed by an arbitrary class of diagram shapes.

Colimits and cocompletions in internal higher category theory

Abstract

We develop a number of basic concepts in the theory of categories internal to an -topos. We discuss adjunctions, limits and colimits as well as Kan extensions for internal categories, and we use these results to prove the universal property of internal presheaf categories. We furthermore construct the free cocompletion of an internal category by colimits that are indexed by an arbitrary class of diagram shapes.
Paper Structure (48 sections, 134 theorems, 152 equations)

This paper contains 48 sections, 134 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Background on $\mathcal{B}$-categories
  3. General conventions and notation
  4. Set-theoretical foundations
  5. $\infty$-topoi
  6. Universe enlargement
  7. Factorisation systems
  8. Simplicial objects, $\mathcal{B}$-categories and $\mathcal{B}$-groupoids
  9. $\mathcal{B}$-categories as sheaves of $\infty$-categories
  10. Objects and morphisms in $\mathcal{B}$-categories
  11. Fully faithful functors and full subcategories
  12. The universe for $\mathcal{B}$-groupoids
  13. Left fibrations and the Grothendieck construction
  14. Slice $\mathcal{B}$-categories and initial objects
  15. Yoneda's lemma
  16. ...and 33 more sections

Key Result

Proposition 2.6.4

The inclusion $\mathop{\mathrm{Cat}}\nolimits(\mathcal{B})\hookrightarrow\mathcal{B}_{\Delta}$ preserves filtered colimits and admits a left adjoint which preserves finite products. Therefore, $\mathop{\mathrm{Cat}}\nolimits(\mathcal{B})$ is presentable and an exponential ideal in $\mathcal{B}_{\Del

Theorems & Definitions (363)

  • Definition 2.6.1: martini2021
  • Remark 2.6.2: martini2021
  • Remark 2.6.3
  • Proposition 2.6.4: martini2021
  • Proposition 2.6.5: martini2021
  • Definition 2.6.6
  • Remark 2.6.7
  • Remark 2.6.8: martini2021
  • Remark 2.6.9: martini2021
  • Proposition 2.7.1: martini2021
  • ...and 353 more