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Fully faithful functors and pushouts of $\infty$-categories

Peter J. Haine, Maxime Ramzi, Jan Steinebrunner

TL;DR

The paper develops a general theory for pushouts in ∞-categories along fully faithful functors, providing explicit formulas for most mapping spaces in the pushout and a necklace-based computation for the remaining fourth mapping anima. It proves stability of fully faithful functors under pushouts and establishes a comprehensive toolkit (via presheaf pullbacks and necklace Segalification) to analyze mapping anima in complex colimits. Key contributions include a detailed lift from 1-categorical Dwyer functors to ∞-categorical pushouts, a Segal-away framework for Segal conditions under pushouts, and a Reedy-extension theorem that yields latching/matching descriptions for functor categories. The results have broad applications to sieve inclusions, Dwyer-type functors, pushout products, and Reedy categories, advancing the understanding of how fully faithful embeddings behave under fundamental colimit constructions in higher category theory.

Abstract

We study stability properties of fully faithful functors, and compute mapping anima in pushouts of $\infty$-categories along fully faithful functors. We provide applications of these calculations to pushouts along Dwyer functors and Reedy categories.

Fully faithful functors and pushouts of $\infty$-categories

TL;DR

The paper develops a general theory for pushouts in ∞-categories along fully faithful functors, providing explicit formulas for most mapping spaces in the pushout and a necklace-based computation for the remaining fourth mapping anima. It proves stability of fully faithful functors under pushouts and establishes a comprehensive toolkit (via presheaf pullbacks and necklace Segalification) to analyze mapping anima in complex colimits. Key contributions include a detailed lift from 1-categorical Dwyer functors to ∞-categorical pushouts, a Segal-away framework for Segal conditions under pushouts, and a Reedy-extension theorem that yields latching/matching descriptions for functor categories. The results have broad applications to sieve inclusions, Dwyer-type functors, pushout products, and Reedy categories, advancing the understanding of how fully faithful embeddings behave under fundamental colimit constructions in higher category theory.

Abstract

We study stability properties of fully faithful functors, and compute mapping anima in pushouts of -categories along fully faithful functors. We provide applications of these calculations to pushouts along Dwyer functors and Reedy categories.

Paper Structure

This paper contains 22 sections, 30 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Linear overview
  4. Notational conventions
  5. Acknowledgments
  6. Characterizations of fully faithful functors
  7. Stability properties of fully faithful functors
  8. Mapping anima in pushouts
  9. The first three types of mapping anima
  10. The fourth mapping anima
  11. Computing pushouts via necklaces
  12. Recollection on necklaces
  13. A pushout in terms of constrained necklaces
  14. Segal conditions away from subsets
  15. Constrained necklaces and fold maps
  16. ...and 7 more sections

Key Result

Theorem 1

Consider a pushout square of \begin{tikzcd}[column sep=2.5em] \Acal \arrow[r, "g"] \arrow[d, "f"', hooked] \arrow[dr, phantom, very near end, "\ulcorner", xshift=0.25em, yshift=-0.25em] & \Ccal \arrow[d, "\fbar"] \\ \Bcal \arrow[r, "\gbar"'] & \Dcal \comma \end{tikzcd Finally, for all $b_0, b_1 \in \Bcal$ we have a pushout square \begin{tikzcd}[column sep = large]

Figures (3)

  • Figure 1: The necklace $N = \Delta^1 \vee \Delta^3 \vee \Delta^2 \vee \Delta^2$, with its joints marked in pink. The alternative notation for this necklace is $([8], \{0,1,4,6,8\})$.
  • Figure 2: Four necklaces that are in $\categ{Nec}_{x,y}^A(X)$, $\categ{Nec}_{x,y}^{[A]}(X)$, $\categ{Nec}_{x,y}^{(A)}(X)$ and $\categ{Nec}_{x,y}(X)$, respectively. (They are chosen such that each of them is not in any of the smaller subcategories.)
  • Figure 3: The left arrow depicts the counit of the colocalization $\categ{Nec}_{x,y}^A(X) \rightleftarrows \categ{Nec}_{x,y}^{[A]}(X)$. The right arrow depicts the unit of the localization $\categ{Nec}_{x,y}^{(A)}(X) \rightleftarrows \categ{Nec}_{x,y}^{[A]}(X)$.

Theorems & Definitions (71)

  • Theorem 1
  • Corollary 2: (\ref{['cor:pushouts_along_Dwyer_functors']})
  • Lemma 1.1: ambidexterity
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.4: (see also arXiv:2103.17141)
  • proof
  • Proposition 2.1
  • ...and 61 more