open import 1Lab.Prelude

module Algebra.Group.Homotopy.BAut where


# Deloopings of automorphism groupsπ

Recall that any set generates a group , given by the automorphisms We also have a generic construction of deloopings: special spaces (for a group where the fundamental group recovers For the specific case of deloping automorphism groups, we can give an alternative construction: The type of small types merely equivalent to has a fundamental group of

module _ {β} (T : Type β) where
BAut : Type (lsuc β)
BAut = Ξ£[ B β Type β ] β₯ T β B β₯

base : BAut
base = T , inc (id , id-equiv)


The first thing we note is that BAut is a connected type, meaning that it only has βa single pointβ, or, more precisely, that all of its interesting information is in its (higher) path spaces:

  connected : (x : BAut) β β₯ x β‘ base β₯
connected = elim! Ξ» b e β inc (p b e) where
p : β b e β (b , inc e) β‘ base
p _ = EquivJ (Ξ» B e β (B , inc e) β‘ base) refl


We now turn to proving that We will define a type family where a value codes for an identification Correspondingly, there are functions to and from these types: The core of the proof is showing that these functions, encode and decode, are inverses.

  Code : BAut β Type β
Code b = T β b .fst

encode : β b β base β‘ b β Code b
encode x p = pathβequiv (ap fst p)

decode : β b β Code b β base β‘ b
decode (b , eqv) rot = Ξ£-pathp (ua rot) (is-propβpathp (Ξ» _ β squash) _ _)


Recall that is the type itself, equipped with the identity equivalence. Hence, to code for an identification it suffices to record β which by univalence corresponds to a path We can not directly extract the equivalence from a given point it is βhidden awayβ under a truncation. But, given an identification we can recover the equivalence by seeing how identifies

  decodeβencode : β b (p : base β‘ b) β decode b (encode b p) β‘ p
decodeβencode b =
J (Ξ» b p β decode b (encode b p) β‘ p)
(Ξ£-prop-square (Ξ» _ β squash) sq)
where
sq : ua (encode base refl) β‘ refl
sq = ap ua pathβequiv-refl β ua-id-equiv


Encode and decode are inverses by a direct application of univalence.

  encodeβdecode : β b (p : Code b) β encode b (decode b p) β‘ p
encodeβdecode b p = ua.Ξ· _


We now have the core result: Specialising encode and decode to the basepoint, we conclude that loop space is equivalent to

  Ξ©ΒΉBAut : (base β‘ base) β (T β T)
(encode base , iso (decode base) (encodeβdecode base) (decodeβencode base))


We can also characterise the h-level of these connected components. Intuitively the h-level should be one more than that of the type weβre delooping, because BAut βonly has one pointβ (itβs connected), and as we established right above, the space of loops of that point is the automorphism group we delooped. The trouble here is that BAut has many points, and while we can pick paths between any two of them, we can not do so continuously (otherwise BAut would be a proposition).

This turns out not to matter! Since βbeing of h-level β is a proposition, our discontinuous (i.e.: truncated) method of picking paths is just excellent. In the case when is contractible, we can directly get at the underlying equivalences, but for the higher h-levels, we proceed exactly by using connectedness.

  BAut-is-hlevel : β n β is-hlevel T n β is-hlevel BAut (1 + n)
BAut-is-hlevel zero hl (x , f) (y , g) = Ξ£-prop-path! (sym (ua f') β ua g')
where
extract : β {X} β is-prop (T β X)
extract f g = ext Ξ» x β ap fst \$
is-contrβis-prop ((f eβ»ΒΉ) .snd .is-eqv (hl .centre))
(f .fst x , is-contrβis-prop hl _ _)
(g .fst x , is-contrβis-prop hl _ _)

f' = β₯-β₯-rec extract (Ξ» x β x) f
g' = β₯-β₯-rec extract (Ξ» x β x) g
BAut-is-hlevel (suc n) hl x y =
β₯-β₯-elimβ {P = Ξ» _ _ β is-hlevel (x β‘ y) (1 + n)}
(Ξ» _ _ β is-hlevel-is-prop _)
(Ξ» p q β transport (apβ (Ξ» a b β is-hlevel (a β‘ b) (1 + n)) (sym p) (sym q))
(Equivβis-hlevel (1 + n) Ξ©ΒΉBAut (β-is-hlevel (1 + n) hl hl)))
(connected x) (connected y)