[Paper Review] Homotopically discrete higher categorical structures

Simona Paoli|arXiv (Cornell University)|May 17, 2016

Homotopy and Cohomology in Algebraic Topology6 references3 citations

TL;DR

This paper introduces homotopically discrete $n$-fold categories as a higher categorical model of 0-types, providing two equivalent descriptions: one via a Segal-type model using iterated nerves, and another via iterated internal equivalence relations. The key contribution is proving that these two descriptions are categorically isomorphic, establishing a foundational model for weakly globular higher categories.

ABSTRACT

We introduce the notion of homotopically discrete n-fold category as an n-fold generalization of a groupoid with no non-trivial loops. We give two equivalent descriptions of this structure: in terms of a Segal-type model and in terms of iterated internal equivalence relations. We also show that homotopically discrete n-fold categories form an n-fold categorical model of 0-types.

Motivation & Objective

• To define and characterize homotopically discrete $n$-fold categories as a higher categorical generalization of groupoids with no non-trivial loops.
• To establish two equivalent descriptions: one based on a Segal-type model using iterated nerves, and another via iterated internal equivalence relations.
• To show that these structures model 0-types in the sense of homotopy theory, generalizing discrete sets in higher dimensions.
• To provide the categorical foundation for weakly globular $n$-fold categories, enabling new models of weak higher categories.

Proposed method

• The definition is constructed inductively: a homotopically discrete $n$-fold category is a simplicial object in homotopically discrete $(n-1)$-fold categories.
• The Segal-type model is built via the nerve functor $J_n: ext{Cat}_{ ext{hd}}^n o [ riangle^{n-1^{ ext{op}}}, ext{Cat}]$, which is levelwise an equivalence relation.
• The second description uses iterated internal equivalence relations in the category of $n$-fold categories, denoted $\mathsf{EqRel}^n$, where each level is an equivalence relation.
• An $n$-equivalence is defined as a higher-dimensional generalization of fully faithful and essentially surjective functors, with a characterization via isomorphisms of discrete models.
• The proof of equivalence between the two descriptions relies on induction on $n$, using the fact that $p^{(n)}X$ and $X_s$ are in $\text{Cat}_{ ext{hd}}^{n-1}$ when $X$ is in $\text{Cat}_{ ext{hd}}^n$, and that $f_{\underline{s}}$ is surjective.
• The isomorphism $\mathsf{EqRel}^n \cong \mathsf{Cat}_{ ext{hd}}^n$ is established by showing that every $X \in \mathsf{Cat}_{ ext{hd}}^n$ arises as $X = (\xi_n X)_0[f_{n}X]$ with $f_{n}X$ surjective, and vice versa.

Experimental results

Research questions

• RQ1 How can the notion of homotopically discrete structure be generalized from groupoids to $n$-fold categories?
• RQ2 What are the two equivalent descriptions of homotopically discrete $n$-fold categories, and how do they relate?
• RQ3 Can homotopically discrete $n$-fold categories serve as a model for 0-types in higher category theory?
• RQ4 How does this structure support the construction of weakly globular $n$-fold categories in subsequent work?

Key findings

• Homotopically discrete $n$-fold categories are shown to be equivalent to $n$-fold categories built via iterated internal equivalence relations, establishing $\mathsf{EqRel}^n \cong \mathsf{Cat}_{\text{hd}}^n$.
• Every homotopically discrete $n$-fold category $X$ is $n$-equivalent to a discrete $n$-fold category $X^d$ via a discretization map $d: X \to X^d$, showing they are homotopically trivial in higher dimensions.
• The nerve functor $J_n$ maps $X \in \mathsf{Cat}_{\text{hd}}^n$ to a levelwise equivalence relation, confirming the Segal-type model structure.
• The induced Segal maps for $X \in \mathsf{Cat}_{\text{hd}}^n$ are $(n-1)$-equivalences, confirming the Segal condition in the higher categorical sense.
• The structure is closed under the $p^{(n)}$-construction: if $X \in \mathsf{Cat}_{\text{hd}}^n$, then $p^{(n)}X \in \mathsf{Cat}_{\text{hd}}^{n-1}$, ensuring consistency across dimensions.
• The characterization of $n$-equivalences via isomorphisms of discrete models provides a practical criterion for equivalence in this category.