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An Enriched Approach to the Strictification of $(\infty,1)$-Categories

Kimball Strong

TL;DR

The paper deconstructs a two-step strictification of $(\infty,1)$-categories to a maximally strict $(\omega,1)$-category, first by locally strictifying hom-objects to $\omega$-groupoids under the Gray tensor, then by globally strictifying the enrichment to a cartesian product. It builds a robust framework of model structures and Quillen adjunctions connecting $\text{sSet}$-Cat, crossed-complex enriched categories and ordinary groupoid enrichment, and identifies explicit generating (a)cyclic cofibrations that encode coherent path lifting. The authors prove conservativity of the strictification in two key regimes—$2$-truncated and $2$-connected $(\infty,1)$-categories—supporting a broader conjecture that the strictification is conservative in general. They also develop an explicit perspective on the global-strictification functor, linking it to linearization phenomena and indecomposables of augmented DGAs, and discuss how these ideas relate to Whitehead-type theorems for higher categories.

Abstract

We define a functor which takes in an $(\infty,1)$-category and outputs an $(ω,1)$-category, the natural maximally "strict" version of an $(\infty,1)$-category. We do this by modeling $(\infty,1)$-categories as categories enriched in $\infty$-groupoids, and then "locally strictifying" (applying the strictification of $\infty$-groupoids to each hom space) to obtain a category enriched in $ω$-groupoids with respect to the Gray tensor product, followed by "globally strictifying" (strictifying the enrichment from the Gray tensor product to the cartesian product) to obtain a category cartesian-enriched in $ω$-groupoids, which is equivalently an $(ω,1)$-category. We conjecture that this functor is conservative, and prove this for two dual special cases: $2$-truncated and $2$-connected $(\infty,1)$-categories. Along the way, we construct a sort of "incoherent walking $(ω,1)$-equivalence," which gives a simpler description of the coherent path lifting condition for fibrations of $(ω,1)$-categories, only involving cells of dimension $\le 3$.

An Enriched Approach to the Strictification of $(\infty,1)$-Categories

TL;DR

The paper deconstructs a two-step strictification of -categories to a maximally strict -category, first by locally strictifying hom-objects to -groupoids under the Gray tensor, then by globally strictifying the enrichment to a cartesian product. It builds a robust framework of model structures and Quillen adjunctions connecting -Cat, crossed-complex enriched categories and ordinary groupoid enrichment, and identifies explicit generating (a)cyclic cofibrations that encode coherent path lifting. The authors prove conservativity of the strictification in two key regimes—-truncated and -connected -categories—supporting a broader conjecture that the strictification is conservative in general. They also develop an explicit perspective on the global-strictification functor, linking it to linearization phenomena and indecomposables of augmented DGAs, and discuss how these ideas relate to Whitehead-type theorems for higher categories.

Abstract

We define a functor which takes in an -category and outputs an -category, the natural maximally "strict" version of an -category. We do this by modeling -categories as categories enriched in -groupoids, and then "locally strictifying" (applying the strictification of -groupoids to each hom space) to obtain a category enriched in -groupoids with respect to the Gray tensor product, followed by "globally strictifying" (strictifying the enrichment from the Gray tensor product to the cartesian product) to obtain a category cartesian-enriched in -groupoids, which is equivalently an -category. We conjecture that this functor is conservative, and prove this for two dual special cases: -truncated and -connected -categories. Along the way, we construct a sort of "incoherent walking -equivalence," which gives a simpler description of the coherent path lifting condition for fibrations of -categories, only involving cells of dimension .

Paper Structure

This paper contains 20 sections, 37 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conventions
  3. Organization of this paper
  4. Background: $(\omega,n)$-categories and crossed complexes
  5. Monoidal structures on Crossed Complexes
  6. The Model Structure on $\text{CrCom}$
  7. Model Structures on Enriched Categories
  8. The Bergner model structure on $\text{sSet}\text{-}\mathcal{Cat}$
  9. The model structure on $\text{CrCom}^\otimes\text{-}\mathcal{Cat}$.
  10. The model structure on $\text{CrCom}^\times\text{-}\mathcal{Cat}$
  11. The model structure on $\text{Gpd}\text{-}\mathcal{Cat}$
  12. The Quillen Adjunctions
  13. The adjunction between $\text{sSet}\text{-}\mathcal{Cat}$ and $\text{CrCom}^\otimes\text{-}\mathcal{Cat}$.
  14. The adjunction between $\text{CrCom}^\otimes\text{-}\mathcal{Cat}$ and $\text{CrCom}^\times\text{-}\mathcal{Cat}$
  15. The adjunctions with $\text{Gpd}\text{-}\mathcal{Cat}$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $f: X \to Y$ be a map of spaces. If then $f$ is a weak equivalence.

Figures (2)

  • Figure 1: Generators of the $\text{CrCom}^\otimes$-category $\tilde{\mathbb{I}}$
  • Figure 2: Generators of the $\text{CrCom}^\times$-category $\tilde{\mathbb{I}}^\times$

Theorems & Definitions (86)

  • Theorem 1.1: Generalized Homological Whitehead Theorem
  • Theorem 1.2: Brown_1981 Whitehead1948
  • Theorem A
  • Theorem B
  • Conjecture
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • ...and 76 more