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Abelian ∞-groups (#1178)
Defines abelian ∞-groups as ∞-groups that are $n$-deloopable for all $n$. In other words, there is a connective spectrum where the group appears as the first type in the sequence.
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‎references.bib

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langid = {hungarian}
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}
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@online{BvDR18,
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@inproceedings{BvDR18,
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title = {Higher {{Groups}} in {{Homotopy Type Theory}}},
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author = {Buchholtz, Ulrik and van Doorn, Floris and Rijke, Egbert},
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date = {2018-02-12},
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eprinttype = {arxiv},
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eprintclass = {cs, math},
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abstract = {We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-L\"of type theory. We investigate ordinary groups from this viewpoint, as well as higher dimensional groups and groups that can be delooped more than once. A major result is the stabilization theorem, which states that if an $n$-type can be delooped $n+2$ times, then it is an infinite loop type. Most of the results have been formalized in the Lean proof assistant.},
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pubstate = {preprint},
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keywords = {03B15,Computer Science - Logic in Computer Science,F.4.1,Mathematics - Algebraic Topology,Mathematics - Logic}
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isbn = {9781450355834},
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doi = {10.1145/3209108.3209150},
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pages = {205--214},
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numpages = {10},
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location = {Oxford, United Kingdom},
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booktitle = {Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science},
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address = {New York, NY, USA},
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publisher = {Association for Computing Machinery},
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series = {LICS '18}
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}
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@article{BW23,

‎src/foundation/connected-types.lagda.md

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A retract-of B →
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is-connected k B →
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is-connected k A
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is-connected-retract-of R c =
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is-contr-retract-of (type-trunc k B) (retract-of-trunc-retract-of R) c
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is-connected-retract-of R =
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is-contr-retract-of (type-trunc k B) (retract-of-trunc-retract-of R)
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```

‎src/higher-group-theory.lagda.md

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```agda
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module higher-group-theory where
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open import higher-group-theory.abelian-higher-groups public
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open import higher-group-theory.cartesian-products-higher-groups public
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open import higher-group-theory.conjugation public
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open import higher-group-theory.cyclic-higher-groups public

‎src/higher-group-theory/abelian-higher-groups.lagda.md

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# Abelian higher groups
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```agda
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module higher-group-theory.abelian-higher-groups where
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```
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<details><summary>Imports</summary>
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```agda
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open import foundation.dependent-pair-types
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open import foundation.equivalences
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open import foundation.small-types
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open import foundation.universe-levels
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open import higher-group-theory.equivalences-higher-groups
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open import higher-group-theory.higher-groups
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open import higher-group-theory.small-higher-groups
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open import structured-types.pointed-equivalences
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open import structured-types.pointed-types
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open import structured-types.small-pointed-types
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open import synthetic-homotopy-theory.connective-spectra
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```
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</details>
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## Idea
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An {{#concept "abelian" Disambiguation="∞-group"}}, or
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{{#concept "commutative" Disambiguation="∞-group"}} ∞-group is a
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[higher group](higher-group-theory.higher-groups.md) `A₀` that is commutative in
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a fully coherent way. There are multiple ways to express this in Homotopy Type
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Theory. One way is to say there is a
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[connective spectrum](synthetic-homotopy-theory.connective-spectra.md) `𝒜` such
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that the ∞-group appears as the first type in the sequence. {{#cite BvDR18}}
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I.e., there exists a sequence of increasingly
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[connected](foundation.connected-types.md) ∞-groups
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```text
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A₀ A₁ A₂ A₃ ⋯ Aᵢ ⋯
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```
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such that
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```text
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Aᵢ ≃∗ Ω Aᵢ₊₁
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```
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Abelian ∞-groups thus give an example of another infinitely coherent structure
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that is definable in Homotopy Type Theory.
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## Definitions
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### The connective spectrum condition of being abelian with respect to a universe level
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```agda
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is-abelian-level-connective-spectrum-condition-∞-Group :
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{l : Level} (l1 : Level) → ∞-Group l → UU (l ⊔ lsuc l1)
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is-abelian-level-connective-spectrum-condition-∞-Group l1 G =
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Σ ( Connective-Spectrum l1)
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( λ A → pointed-type-∞-Group G ≃∗ pointed-type-Connective-Spectrum A 0)
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```
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### The connective spectrum condition of being abelian
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```agda
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is-abelian-connective-spectrum-condition-∞-Group :
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{l : Level} → ∞-Group l → UU (lsuc l)
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is-abelian-connective-spectrum-condition-∞-Group {l} G =
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is-abelian-level-connective-spectrum-condition-∞-Group l G
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```
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## References
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{{#bibliography}}
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## External links
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- [abelian infinity-group](https://ncatlab.org/nlab/show/abelian+infinity-group)
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at $n$Lab

‎src/synthetic-homotopy-theory.lagda.md

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open import synthetic-homotopy-theory.coforks-cocones-under-sequential-diagrams public
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open import synthetic-homotopy-theory.conjugation-loops public
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open import synthetic-homotopy-theory.connected-set-bundles-circle public
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open import synthetic-homotopy-theory.connective-prespectra public
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open import synthetic-homotopy-theory.connective-spectra public
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open import synthetic-homotopy-theory.dependent-cocones-under-sequential-diagrams public
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open import synthetic-homotopy-theory.dependent-cocones-under-spans public
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open import synthetic-homotopy-theory.dependent-coforks public

‎src/synthetic-homotopy-theory/connective-prespectra.lagda.md

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# Connective prespectra
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```agda
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module synthetic-homotopy-theory.connective-prespectra where
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```
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<details><summary>Imports</summary>
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```agda
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open import elementary-number-theory.natural-numbers
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open import foundation.connected-types
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open import foundation.dependent-pair-types
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open import foundation.equivalences
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open import foundation.function-types
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open import foundation.identity-types
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open import foundation.propositions
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open import foundation.truncation-levels
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open import foundation.universe-levels
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open import structured-types.pointed-equivalences
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open import structured-types.pointed-maps
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open import structured-types.pointed-types
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open import synthetic-homotopy-theory.loop-spaces
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open import synthetic-homotopy-theory.prespectra
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open import synthetic-homotopy-theory.suspensions-of-pointed-types
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open import synthetic-homotopy-theory.suspensions-of-types
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```
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</details>
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## Idea
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A [prespectrum](synthetic-homotopy-theory.prespectra.md) is
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{{#concept "connective" Disambiguation="prespectrum" Agda=is-connective-Prespectrum}}
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if the $n$th type in the [sequence](foundation.sequences.md) is
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$n$-[connected](foundation.connected-types.md).
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### The predicate on prespectra of being connective
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```agda
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module _
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{l : Level} (A : Prespectrum l)
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where
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is-connective-Prespectrum : UU l
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is-connective-Prespectrum =
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(n : ℕ) → is-connected (truncation-level-ℕ n) (type-Prespectrum A n)
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is-prop-is-connective-Prespectrum : is-prop is-connective-Prespectrum
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is-prop-is-connective-Prespectrum =
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is-prop-Π
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( λ n →
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is-prop-is-connected (truncation-level-ℕ n) (type-Prespectrum A n))
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is-connective-prop-Prespectrum : Prop l
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is-connective-prop-Prespectrum =
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is-connective-Prespectrum , is-prop-is-connective-Prespectrum
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```
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### The type of connective prespectra
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```agda
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Connective-Prespectrum : (l : Level) → UU (lsuc l)
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Connective-Prespectrum l = Σ (Prespectrum l) (is-connective-Prespectrum)
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module _
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{l : Level} (A : Connective-Prespectrum l)
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where
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prespectrum-Connective-Prespectrum : Prespectrum l
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prespectrum-Connective-Prespectrum = pr1 A
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pointed-type-Connective-Prespectrum : ℕ → Pointed-Type l
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pointed-type-Connective-Prespectrum =
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pointed-type-Prespectrum prespectrum-Connective-Prespectrum
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type-Connective-Prespectrum : ℕ → UU l
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type-Connective-Prespectrum =
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type-Prespectrum prespectrum-Connective-Prespectrum
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point-Connective-Prespectrum : (n : ℕ) → type-Connective-Prespectrum n
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point-Connective-Prespectrum =
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point-Prespectrum prespectrum-Connective-Prespectrum
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pointed-adjoint-structure-map-Connective-Prespectrum :
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(n : ℕ) →
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pointed-type-Connective-Prespectrum n →∗
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Ω (pointed-type-Connective-Prespectrum (succ-ℕ n))
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pointed-adjoint-structure-map-Connective-Prespectrum =
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pointed-adjoint-structure-map-Prespectrum prespectrum-Connective-Prespectrum
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adjoint-structure-map-Connective-Prespectrum :
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(n : ℕ) →
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type-Connective-Prespectrum n →
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type-Ω (pointed-type-Connective-Prespectrum (succ-ℕ n))
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adjoint-structure-map-Connective-Prespectrum =
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adjoint-structure-map-Prespectrum prespectrum-Connective-Prespectrum
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preserves-point-adjoint-structure-map-Connective-Prespectrum :
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(n : ℕ) →
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adjoint-structure-map-Prespectrum (pr1 A) n (point-Prespectrum (pr1 A) n) ＝
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refl-Ω (pointed-type-Prespectrum (pr1 A) (succ-ℕ n))
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preserves-point-adjoint-structure-map-Connective-Prespectrum =
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preserves-point-adjoint-structure-map-Prespectrum
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prespectrum-Connective-Prespectrum
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is-connective-Connective-Prespectrum :
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is-connective-Prespectrum prespectrum-Connective-Prespectrum
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is-connective-Connective-Prespectrum = pr2 A
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```
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### The structure maps of a connective prespectrum
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```agda
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module _
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{l : Level} (A : Connective-Prespectrum l) (n : ℕ)
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where
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pointed-structure-map-Connective-Prespectrum :
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suspension-Pointed-Type (pointed-type-Connective-Prespectrum A n) →∗
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pointed-type-Connective-Prespectrum A (succ-ℕ n)
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pointed-structure-map-Connective-Prespectrum =
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pointed-structure-map-Prespectrum (prespectrum-Connective-Prespectrum A) n
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structure-map-Connective-Prespectrum :
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suspension (type-Connective-Prespectrum A n) →
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type-Connective-Prespectrum A (succ-ℕ n)
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structure-map-Connective-Prespectrum =
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map-pointed-map pointed-structure-map-Connective-Prespectrum
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preserves-point-structure-map-Connective-Prespectrum :
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structure-map-Connective-Prespectrum north-suspension ＝
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point-Connective-Prespectrum A (succ-ℕ n)
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preserves-point-structure-map-Connective-Prespectrum =
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preserves-point-pointed-map pointed-structure-map-Connective-Prespectrum
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```
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## External links
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- [connective spectrum](https://ncatlab.org/nlab/show/connective+spectrum) at
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$n$Lab

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