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The presentable stable envelope of an exact category

Marius Nielsen, Christoph Winges

Abstract

We prove an analogue of the Gabriel--Quillen embedding theorem for exact $\infty$-categories, giving rise to a presentable version of Klemenc's stable envelope of an exact $\infty$-category. Moreover, we construct a symmetric monoidal structure on the $\infty$-category of small exact $\infty$-categories and discuss the multiplicative properties of the Gabriel--Quillen embedding. For $E$ an Adams-type homotopy associative ring spectrum, this allows us to identify the symmetric monoidal $\infty$-category of $E$-based synthetic spectra with the presentable stable envelope of the exact $\infty$-category of compact spectra with finite projective $E$-homology. In addition, we show that algebraic K-theory, considered as a functor on exact $\infty$-categories, admits a unique delooping as a localising invariant.

The presentable stable envelope of an exact category

Abstract

We prove an analogue of the Gabriel--Quillen embedding theorem for exact -categories, giving rise to a presentable version of Klemenc's stable envelope of an exact -category. Moreover, we construct a symmetric monoidal structure on the -category of small exact -categories and discuss the multiplicative properties of the Gabriel--Quillen embedding. For an Adams-type homotopy associative ring spectrum, this allows us to identify the symmetric monoidal -category of -based synthetic spectra with the presentable stable envelope of the exact -category of compact spectra with finite projective -homology. In addition, we show that algebraic K-theory, considered as a functor on exact -categories, admits a unique delooping as a localising invariant.

Paper Structure

This paper contains 5 sections, 39 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. The presentable Gabriel--Quillen embedding
  3. Tensor products of exact categories
  4. The example of synthetic spectra
  5. Localising invariants

Key Result

Theorem 1.1

Let $\mathcal{C}$ be a small exact $\infty$-category. Then the Yoneda embedding $\mathcal{C} \to \mathcal{P}_\mathrm{lex}(\mathcal{C})$ exhibits $\mathcal{C}$ as an extension-closed subcategory of a Grothendieck prestable $\infty$-category. Furthermore, for every pointed cocomplete $\infty$-category Here $\mathcal{P}_\mathrm{lex}(\mathcal{C})\subseteq \mathcal{P}(\mathcal{C})$ denotes the full sub

Theorems & Definitions (100)

  • Theorem 1.1: Gabriel--Quillen embedding theorem, \ref{['thm:plex', 'thm:gabriel-quillen']}
  • Definition 1.2
  • Theorem 1.3: \ref{['cor:exact-symm-monoidal']}
  • Theorem 1.4: \ref{['syn:E-based-synthetic-exact']}
  • Remark 1.5
  • Proposition 1.6: \ref{['cor:nc-K-universal-property']}
  • Proposition 1.7: Gillet--Waldhausen, \ref{['cor:gillet-waldhausen']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 90 more