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Central H-spaces and banded types

Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxerås Flaten, Egbert Rijke

TL;DR

This work develops a HoTT-based theory of central H-spaces and their deloopings via banded structures. It introduces central types, proves they admit a unique infinite delooping obtained from banded self-equivalences $\mathrm{BAut}_{1}(A)$, and equips the band space with an abelian H-space structure through tensoring of bands. The paper also reformulates deloopings in terms of $A$-torsors and demonstrates how centrality yields infinite deloopings for $A$ and all pointed self-maps, with concrete constructions for Eilenberg–Mac Lane spaces $\operatorname{K}(G,n)$ and $G$-torsors $TG$. It further analyzes the moduli of H-space structures, rules out H-space structures on even spheres, and provides stable, two-homotopy-group classifications for central, truncated types, illustrating a robust HoTT framework for infinite loop space theory and GEM decompositions. Overall, it establishes a cohesive, intrinsic approach to centrality, banded deloopings, and torsor-based descriptions that generalize classical GEM results to arbitrary $\infty$-topoi.

Abstract

We introduce and study central types, which are generalizations of Eilenberg-Mac Lane spaces. A type is central when it is equivalent to the component of the identity among its own self-equivalences. From centrality alone we construct an infinite delooping in terms of a tensor product of banded types, which are the appropriate notion of torsor for a central type. Our constructions are carried out in homotopy type theory, and therefore hold in any $\infty$-topos. Even when interpreted into the $\infty$-topos of spaces, our approach to constructing these deloopings is new. Along the way, we further develop the theory of H-spaces in homotopy type theory, including their relation to evaluation fibrations and Whitehead products. These considerations let us, for example, rule out the existence of H-space structures on the $2n$-sphere for $n > 0$. We also give a novel description of the moduli space of H-space structures on an H-space. Using this description, we generalize a formula of Arkowitz-Curjel and Copeland for counting the number of path components of this moduli space. As an application, we deduce that the moduli space of H-space structures on the $3$-sphere is $Ω^6 \mathbb{S}^3$.

Central H-spaces and banded types

TL;DR

This work develops a HoTT-based theory of central H-spaces and their deloopings via banded structures. It introduces central types, proves they admit a unique infinite delooping obtained from banded self-equivalences , and equips the band space with an abelian H-space structure through tensoring of bands. The paper also reformulates deloopings in terms of -torsors and demonstrates how centrality yields infinite deloopings for and all pointed self-maps, with concrete constructions for Eilenberg–Mac Lane spaces and -torsors . It further analyzes the moduli of H-space structures, rules out H-space structures on even spheres, and provides stable, two-homotopy-group classifications for central, truncated types, illustrating a robust HoTT framework for infinite loop space theory and GEM decompositions. Overall, it establishes a cohesive, intrinsic approach to centrality, banded deloopings, and torsor-based descriptions that generalize classical GEM results to arbitrary -topoi.

Abstract

We introduce and study central types, which are generalizations of Eilenberg-Mac Lane spaces. A type is central when it is equivalent to the component of the identity among its own self-equivalences. From centrality alone we construct an infinite delooping in terms of a tensor product of banded types, which are the appropriate notion of torsor for a central type. Our constructions are carried out in homotopy type theory, and therefore hold in any -topos. Even when interpreted into the -topos of spaces, our approach to constructing these deloopings is new. Along the way, we further develop the theory of H-spaces in homotopy type theory, including their relation to evaluation fibrations and Whitehead products. These considerations let us, for example, rule out the existence of H-space structures on the -sphere for . We also give a novel description of the moduli space of H-space structures on an H-space. Using this description, we generalize a formula of Arkowitz-Curjel and Copeland for counting the number of path components of this moduli space. As an application, we deduce that the moduli space of H-space structures on the -sphere is .
Paper Structure (16 sections, 58 theorems, 50 equations)

This paper contains 16 sections, 58 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. H-spaces and evaluation fibrations
  3. H-space structures
  4. (α,β)-extensions and Whitehead products
  5. Evaluation fibrations
  6. Unique H-space structures
  7. Central types
  8. Bands and torsors
  9. Types banded by a central type
  10. Tensoring bands
  11. Bands and torsors
  12. Examples and non-examples
  13. The H-space of G-torsors
  14. Eilenberg--Mac Lane spaces
  15. Products of Eilenberg--Mac Lane spaces
  16. ...and 1 more sections

Key Result

Proposition 2.2

Let $A$ be a pointed type. The following types are equivalent: Moreover, the type of non-coherent H-space structures on $A$ is equivalent to the type of families $\mu : \prod_{a:A} (A, \mathrm{{pt}}) \to_* (A,a)$ of pointed maps equipped with an unpointed homotopy $\mu(\mathrm{{pt}}) = \mathrm{{id}}_A$.

Theorems & Definitions (141)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 131 more