fact stringlengths 4 2.87k | type stringclasses 17 values | library stringclasses 91 values | imports listlengths 0 19 | filename stringclasses 497 values | symbolic_name stringlengths 1 75 | docstring stringlengths 5 3.99k ⌀ |
|---|---|---|---|---|---|---|
isqidem_qidem {X : Type} (f : QuasiIdempotent X)
: IsQuasiIdempotent f
:= f.2. | Instance | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | isqidem_qidem | null |
isqidem_idmap (X : Type) : @IsQuasiIdempotent X idmap _
:= fun _ => 1. | Instance | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | isqidem_idmap | The identity function has a canonical structure of a quasi-idempotent. |
qidem_idmap (X : Type) : QuasiIdempotent X. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | qidem_idmap | null |
ispreidem_retract {X : Type} (R : RetractOf X)
: IsPreIdempotent (retract_idem R). | Instance | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | ispreidem_retract | First we show that given a retract, the composite [s o r] is quasi-idempotent. |
preidem_retract {X : Type} (R : RetractOf X)
: PreIdempotent X
:= (retract_idem R ; ispreidem_retract R). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | preidem_retract | null |
isqidem_retract {X : Type} (R : RetractOf X)
: IsQuasiIdempotent (retract_idem R). | Instance | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | isqidem_retract | null |
qidem_retract {X : Type} (R : RetractOf X)
: QuasiIdempotent X
:= (preidem_retract R ; isqidem_retract R). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | qidem_retract | null |
ispreidem_split {X : Type} (f : X -> X) (S : Splitting f)
: IsPreIdempotent f. | Instance | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | ispreidem_split | In particular, it follows that any split function is quasi-idempotent. |
isqidem_split {X : Type} (f : X -> X) (S : Splitting f)
: @IsQuasiIdempotent X f (ispreidem_split f S). | Instance | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | isqidem_split | null |
I := isidem f. | Let | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | I | null |
J : forall x, ap f (I x) = I (f x)
:= isidem2 f. | Let | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | J | null |
split_idem : Type
:= { a : nat -> X & forall n, f (a n.+1) = a n }. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem | The splitting will be the sequential limit of the sequence [... -> X -> X -> X]. |
split_idem_pr1 : split_idem -> (nat -> X)
:= pr1. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_pr1 | null |
split_idem_sect : split_idem -> X
:= fun a => a 0. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_sect | The section, retraction, and the fact that the composite in one direction is [f] are easy. |
split_idem_retr : X -> split_idem. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_retr | null |
split_idem_splits (x : X)
: split_idem_sect (split_idem_retr x) = f x
:= 1. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_splits | null |
path_split_idem {a a' : split_idem}
(p : a.1 == a'.1)
(q : forall n, a.2 n @ p n = ap f (p n.+1) @ a'.2 n)
: a = a'. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | path_split_idem | What remains is to show that the composite in the other direction is the identity. We begin by showing how to construct paths in [split_idem]. |
sect_path_split_idem {a a' : split_idem}
(p : a.1 == a'.1)
(q : forall n, a.2 n @ p n = ap f (p n.+1) @ a'.2 n)
: ap split_idem_sect (path_split_idem p q) = p 0. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | sect_path_split_idem | And we verify how those paths compute under [split_idem_sect]. |
nudge (a : split_idem) : split_idem. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | nudge | Next we show that every element of [split_idem] can be nudged to an equivalent one in which all the elements of [X] occurring are double applications of [f]. |
nudge_eq a : nudge a = a. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | nudge_eq | null |
split_idem_issect_part1 (a : split_idem) (n : nat)
: f (f (a n.+1)) = f (a 0). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_issect_part1 | Now we're ready to prove the final condition. We prove the two arguments of [path_split_idem] separately, in order to make the first one transparent and the second opaque. |
split_idem_issect_part2 (a : split_idem) (n : nat)
: ap f (ap f (a.2 n.+1)) @ split_idem_issect_part1 a n =
ap f ((ap f (a.2 n.+1) @ (I (a.1 n.+1))^) @ split_idem_issect_part1 a n) @ I (a.1 0). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_issect_part2 | null |
split_idem_issect (a : split_idem)
: split_idem_retr (split_idem_sect a) = a. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_issect | null |
split_idem_retract : RetractOf X
:= Build_RetractOf
X split_idem split_idem_retr split_idem_sect split_idem_issect. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_retract | null |
split_idem_split : Splitting f
:= (split_idem_retract ; split_idem_splits). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_split | null |
split_idem_preidem (x : X)
: ap split_idem_sect (split_idem_issect (split_idem_retr x))
= I x. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_preidem | We end this section by showing that we can recover the witness [I] of pre-idempotence from the splitting. |
split_idem_retract' `{fs : Funext} {X : Type}
: QuasiIdempotent X -> RetractOf X
:= fun f => split_idem_retract f. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_retract | null |
split_idem_split' `{fs : Funext} {X : Type}
(f : QuasiIdempotent X)
: Splitting f
:= split_idem_split f. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | split_idem_split | null |
A := retract_type R. | Let | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | A | null |
r := retract_retr R. | Let | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | r | null |
s := retract_sect R. | Let | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | s | null |
H := retract_issect R. | Let | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | H | null |
equiv_split_idem_retract_isadj (a : split_idem (s o r))
: H (r (s (r (split_idem_sect (s o r) a)))) @
H (r (split_idem_sect (s o r) a)) =
ap (r o split_idem_sect (s o r))
(ap (split_idem_retr (s o r))
(1 @
ap (split_idem_sect (s o r))
(split_idem_issect (s o r) a)) @
split_idem_issect (s o r) a). | Lemma | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | equiv_split_idem_retract_isadj | We begin by constructing an equivalence between [split_idem (s o r)] and [A]. We want to make this equivalence transparent so that we can reason about it later. In fact, we want to reason not only about the equivalence function and its inverse, but the section and retraction homotopies! Therefore, instead of using [equiv_adjointify] we will give the coherence proof explicitly, so that we can control these homotopies. However, we can (and should) make the coherence proof itself opaque. Thus, we prove it first, and end it with [Qed]. |
equiv_split_idem_retract
: split_idem (s o r) <~> A. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | equiv_split_idem_retract | Now we can construct the desired equivalence. |
equiv_split_idem_retract_retr (x : X)
: equiv_split_idem_retract (split_idem_retr (s o r) x) = r x
:= H (r x). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | equiv_split_idem_retract_retr | It is easy to show that this equivalence respects the section and the retraction. |
equiv_split_idem_retract_sect (a : A)
: split_idem_sect (s o r) (equiv_split_idem_retract^-1 a) = s a
:= ap s (H a). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | equiv_split_idem_retract_sect | null |
equiv_split_idem_retract_issect (a : A)
: ap equiv_split_idem_retract
(split_idem_issect (s o r) (equiv_split_idem_retract^-1 a))
@ eisretr equiv_split_idem_retract a
= equiv_split_idem_retract_retr
(split_idem_sect (s o r) (equiv_split_idem_retract^-1 a))
@ ap r (equiv_split_idem_retract_sect a)
@ H a. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | equiv_split_idem_retract_issect | Less trivial is to show that it respects the retract homotopy. |
equiv_split_idem_retract_splits (x : X)
: split_idem_splits (s o r) x
= ap (split_idem_sect (s o r))
(eissect equiv_split_idem_retract
(split_idem_retr (s o r) x))^
@ equiv_split_idem_retract_sect
(equiv_split_idem_retract (split_idem_retr (s o r) x))
@ ap s (equiv_split_idem_retract_retr x). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | equiv_split_idem_retract_splits | We will also show that it respects the homotopy to the split map. It's unclear whether this has any use. |
retract_retractof_qidem : RetractOf (QuasiIdempotent X). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | retract_retractof_qidem | null |
splitting_retractof_isqidem (f : X -> X)
: RetractOf { I : IsPreIdempotent f & IsQuasiIdempotent f }. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | splitting_retractof_isqidem | We have a similar result for splittings of a fixed map [f]. |
Splitting_PreIdempotent (f : PreIdempotent X)
:= { S : Splitting f &
forall x, ap f (S.2 x)^
@ (S.2 (retract_idem S.1 x))^
@ ap (retract_sect S.1) (retract_issect S.1 (retract_retr S.1 x))
@ S.2 x
= (isidem f x) }. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | Splitting_PreIdempotent | And also for splittings of a fixed map that also induce a given witness of pre-idempotency. |
splitting_preidem_retractof_qidem (f : PreIdempotent X)
: RetractOf (IsQuasiIdempotent f). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | splitting_preidem_retractof_qidem | null |
IsIdempotent {X : Type} (f : X -> X)
:= is_coherent_idem : split_idem (retract_idem (splitting_retractof_isqidem f)). | Class | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | IsIdempotent | null |
Build_IsIdempotent {X : Type} (f : X -> X)
: Splitting f -> IsIdempotent f
:= (equiv_split_idem_retract (splitting_retractof_isqidem f))^-1. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | Build_IsIdempotent | null |
isidem_isqidem {X : Type} (f : X -> X) `{IsQuasiIdempotent _ f}
: IsIdempotent f
:= Build_IsIdempotent f (split_idem_split f). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | isidem_isqidem | null |
#[export] Instance ispreidem_isidem {X : Type} (f : X -> X)
`{IsIdempotent _ f} : IsPreIdempotent f. | Instance | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | ispreidem_isidem | null |
#[export] Instance isqidem_isidem {X : Type} (f : X -> X)
`{IsIdempotent _ f} : @IsQuasiIdempotent X f (ispreidem_isidem f). | Instance | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | isqidem_isidem | null |
Idempotent (X : Type) := { f : X -> X & IsIdempotent f }. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | Idempotent | null |
idempotent_pr1 {X : Type} : Idempotent X -> (X -> X) := pr1. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | idempotent_pr1 | null |
#[export] Instance isidem_idem (X : Type) (f : Idempotent X) : IsIdempotent f
:= f.2. | Instance | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | isidem_idem | null |
equiv_idempotent_retractof (X : Type)
: Idempotent X <~> RetractOf X. | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | equiv_idempotent_retractof | The above definitions depend on [Univalence]. Technically this is the case by their construction, since they are a splitting of a map that we only know to be idempotent in the presence of univalence. This map could be defined, and hence "split", without univalence; but also only with univalence do we know that they have the right homotopy type. Thus univalence is used in two places: concluding (meta-theoretically) from HTT 4.4.5.14 that [RetractOf X] has the right homotopy type, and showing (in the next lemma) that it is equivalent to [Idempotent X]. In the absence of univalence, we don't currently have *any* provably-correct definition of the type of coherent idempotents; it ought to involve an infinite tower of coherences as defined in HTT section 4.4.5. However, there may be some Yoneda-like meta-theoretic argument which would imply that the above-defined types do have the correct homotopy type without univalence (though almost certainly not without funext). |
#[export] Instance isidem_idmap (X : Type@{i})
: @IsIdempotent@{i i j} X idmap
:= Build_IsIdempotent idmap (splitting_idmap X). | Instance | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | isidem_idmap | For instance, here is the standard coherent idempotent structure on the identity map. |
idem_idmap (X : Type@{i}) : Idempotent@{i i j} X
:= (idmap ; isidem_idmap X). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | idem_idmap | null |
contr_splitting_preidem_idmap {ua : Univalence} (X : Type)
: Contr (Splitting_PreIdempotent (preidem_idmap X)). | Definition | Root | [
"HoTT",
"HFiber",
"Constant",
"Truncations",
"Homotopy"
] | theories/Idempotents.v | contr_splitting_preidem_idmap | We have shown that every quasi-idempotent can be "coherentified" into a fully coherent idempotent, analogously to how every quasi-inverse can be coherentified into an equivalence. However, just as for quasi-inverses, not every witness to quasi-idempotency *is itself* coherent. This is in contrast to a witness of pre-idempotency, which (if it extends to a quasi-idempotent) can itself be extended to a coherent idempotent; this is roughly the content of [split_idem_preidem] and [splitting_preidem_retractof_qidem].
The key step in showing this is to observe that when [f] is the identity, the retract type [Splitting_PreIdempotent f] of [splitting_preidem_retractof_qidem] is equivalent to the type of types-equivalent-to-[X], and hence contractible. |
IsOProjective (O : Modality) (X : Type) : Type
:= forall A, In O A -> forall B, In O B ->
forall f : X -> B, forall p : A -> B,
IsSurjection p -> merely (exists s : X -> A, p o s == f). | Definition | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | IsOProjective | To quantify over all truncation levels including infinity, we parametrize [IsOProjective] by a [Modality]. When specializing to [IsOProjective purely] we get an (oo,-1)-projectivity predicate, [IsProjective]. When specializing to [IsOProjective (Tr n)] we get an (n,-1)-projectivity predicate, [IsTrProjective]. |
iff_isoprojective_surjections_split
(O : Modality) (X : Type) `{In O X}
: IsOProjective O X <->
(forall (Y : Type), In O Y -> forall (p : Y -> X),
IsSurjection p -> merely (exists s : X -> Y, p o s == idmap)). | Proposition | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | iff_isoprojective_surjections_split | A type X is projective if and only if surjections into X merely split. |
equiv_isoprojective_surjections_split
`{Funext} (O : Modality) (X : Type) `{In O X}
: IsOProjective O X <~>
(forall (Y : Type), In O Y -> forall (p : Y -> X),
IsSurjection p -> merely (exists s : X -> Y, p o s == idmap)). | Corollary | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | equiv_isoprojective_surjections_split | null |
HasOChoice (O : Modality) (A : Type) :=
hasochoice :
forall (B : A -> Type), (forall x, In O (B x)) ->
(forall x, merely (B x)) -> merely (forall x, B x). | Class | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | HasOChoice | In topos theory, an object X is said to be projective if the axiom of choice holds when making choices indexed by X. We will refer to this as [HasOChoice], to avoid confusion with [IsOProjective] above. In similarity with [IsOProjective], we parametrize it by a [Modality]. |
hasochoice_sigma
`{Funext} {A : Type} {B : A -> Type} (O : Modality)
(chA : HasOChoice O A)
(chB : forall a : A, HasOChoice O (B a))
: HasOChoice O {a : A | B a}. | Instance | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | hasochoice_sigma | null |
isoprojective_ochoice (O : Modality) (X : Type)
: HasOChoice O X -> IsOProjective O X. | Proposition | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | isoprojective_ochoice | null |
hasochoice_oprojective (O : Modality) (X : Type) `{In O X}
: IsOProjective O X -> HasOChoice O X. | Proposition | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | hasochoice_oprojective | null |
iff_isoprojective_hasochoice (O : Modality) (X : Type) `{In O X}
: IsOProjective O X <-> HasOChoice O X. | Proposition | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | iff_isoprojective_hasochoice | null |
equiv_isoprojective_hasochoice
`{Funext} (O : Modality) (X : Type) `{In O X}
: IsOProjective O X <~> HasOChoice O X. | Proposition | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | equiv_isoprojective_hasochoice | null |
isprojective_unit : IsProjective Unit. | Proposition | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | isprojective_unit | null |
projective_cover_AC `{Univalence} (A : Type)
: merely (exists X:HSet, exists p : X -> A, IsSurjection p). | Proposition | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | projective_cover_AC | (Exercise 7.9) Assuming AC_(oo,-1) every type merely has a projective cover. |
equiv_isprojective_ishset_AC `{Univalence} (X : Type)
: IsProjective X <~> IsHSet X. | Theorem | Root | [
"Basics",
"Types",
"Truncations",
"Modalities",
"Limits"
] | theories/Projective.v | equiv_isprojective_ishset_AC | Assuming AC_(oo,-1), projective types are exactly sets. |
path_forall_1_beta `{Funext} A B x P f g e Px
: @transport (forall a : A, B a) (fun f => P (f x)) f g (@path_forall _ _ _ _ _ e) Px
= @transport (B x) P (f x) (g x) (e x) Px. | Lemma | Root | [
"Basics",
"Types",
"Tactics"
] | theories/Tactics.v | path_forall_1_beta | The basic idea is expressed in the type of this lemma. |
path_forall_recr_beta' `{Funext} A B x0 P f g e Px
: @transport (forall a : A, B a)
(fun f => P f (f x0))
f
g
(@path_forall _ _ _ _ _ e)
Px
= @transport ((forall a, B a) * B x0)
(fun x => P (fst x) (snd x))
(f, f x0)
(g, g x0)
(path_prod' (@path_forall _ _ _ _ _ e) (e x0))
Px. | Lemma | Root | [
"Basics",
"Types",
"Tactics"
] | theories/Tactics.v | path_forall_recr_beta | The powerful recursive case |
path_forall_recr_beta `{Funext} A B x0 P f g e Px
: @transport (forall a : A, B a)
(fun f => P f (f x0))
f
g
(@path_forall _ _ _ _ _ e)
Px
= @transport (forall x : A, B x)
(fun x => P x (g x0))
f
g
(@path_forall H A B f g e)
(@transport (B x0)
(fun y => P f y)
(f x0)
(g x0)
(e x0)
Px). | Lemma | Root | [
"Basics",
"Types",
"Tactics"
] | theories/Tactics.v | path_forall_recr_beta | Rewrite the recursive case after clean-up |
path_forall_2_beta' `{Funext} A B x0 x1 P f g e Px
: @transport (forall a : A, B a) (fun f => P (f x0) (f x1)) f g (@path_forall _ _ _ _ _ e) Px
= @transport (B x0 * B x1) (fun x => P (fst x) (snd x)) (f x0, f x1) (g x0, g x1) (path_prod' (e x0) (e x1)) Px. | Lemma | Root | [
"Basics",
"Types",
"Tactics"
] | theories/Tactics.v | path_forall_2_beta | An example showing that it works |
path_forall_2_beta `{Funext} A B x0 x1 P f g e Px
: @transport (forall a : A, B a) (fun f => P (f x0) (f x1)) f g (@path_forall _ _ _ _ _ e) Px
= transport (fun y : B x1 => P (g x0) y) (e x1)
(transport (fun y : B x0 => P y (f x1)) (e x0) Px). | Lemma | Root | [
"Basics",
"Types",
"Tactics"
] | theories/Tactics.v | path_forall_2_beta | null |
match_eta {T} {x y : T} (H0 : x = y)
: (H0 = match H0 in (_ = y) return (x = y) with
| idpath => idpath
end)
:= match H0 with idpath => idpath end. | Definition | Root | [
"Basics",
"Types",
"Tactics"
] | theories/Tactics.v | match_eta | Now some lemmas about trivial [match]es |
match_eta1 {T} {x : T} (E : x = x)
: (match E in (_ = y) return (x = y) with
| idpath => idpath
end = idpath)
-> idpath = E
:= fun H => ((H # match_eta E) ^)%path. | Definition | Root | [
"Basics",
"Types",
"Tactics"
] | theories/Tactics.v | match_eta1 | null |
match_eta2 {T} {x : T} (E : x = x)
: (idpath
= match E in (_ = y) return (x = y) with
| idpath => idpath
end)
-> idpath = E
:= fun H => match_eta1 E (H ^)%path. | Definition | Root | [
"Basics",
"Types",
"Tactics"
] | theories/Tactics.v | match_eta2 | null |
internal_paths_rew_r_to_transport {A : Type} {x y : A} (P : A -> Type) (u : P y) (p : x = y)
: internal_paths_rew_r P u p = transport P p^ u. | Definition | Root | [
"Basics",
"Types",
"Tactics"
] | theories/Tactics.v | internal_paths_rew_r_to_transport | Unfortunately, the more common [rewrite ->] uses [internal_paths_rew_r], which is not definitionally equal to something involving [transport]. However, we do have a propositional equality. The arguments here match the arguments that [internal_paths_rew_r] takes. |
Aut (X : Type) : ooGroup
:= Build_ooGroup [BAut X, _] _. | Definition | Algebra | [
"Basics",
"Truncations",
"Algebra",
"Universes",
"Pointed"
] | theories/Algebra/Aut.v | Aut | We define [Aut X] using the pointed, connected type [BAut X]. |
IsCongruence {G} `{SgOp G} (R : Relation G) := {
iscong {x x' y y'} : R x x' -> R y y' -> R (x * y) (x' * y');
}. | Class | Algebra | [
"Classes"
] | theories/Algebra/Congruence.v | IsCongruence | null |
ooAction (G : ooGroup)
:= classifying_space G -> Type. | Definition | Algebra | [
"Basics",
"Algebra"
] | theories/Algebra/ooAction.v | ooAction | * Actions of oo-Groups |
action_space {G} : ooAction G -> Type
:= fun X => X (point _). | Definition | Algebra | [
"Basics",
"Algebra"
] | theories/Algebra/ooAction.v | action_space | null |
ooGroup :=
{ classifying_space : pType ;
isconn_classifying_space :: IsConnected 0 classifying_space
}. | Record | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | ooGroup | ** Definition |
group_type (G : ooGroup) : Type
:= point (B G) = point (B G). | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | group_type | null |
group_loops (X : pType)
: ooGroup. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | group_loops | Every pointed type has a loop space that is an oo-group. |
loops_group (X : pType)
: loops X <~> group_loops X. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | loops_group | Unfortunately, the underlying type of that oo-group is not *definitionally* the same as the ordinary loop space, but it is equivalent to it. |
ooGroupHom (G H : ooGroup)
:= B G ->* B H. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | ooGroupHom | *** Definition |
grouphom_fun {G H} (phi : ooGroupHom G H) : G -> H
:= fmap loops phi. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | grouphom_fun | null |
group_loops_functor
{X Y : pType} (f : X ->* Y)
: ooGroupHom (group_loops X) (group_loops Y). | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | group_loops_functor | The loop group functor takes values in oo-group homomorphisms. |
loops_functor_group
{X Y : pType} (f : X ->* Y)
: fmap loops (group_loops_functor f) o loops_group X
== loops_group Y o fmap loops f. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | loops_functor_group | And this functor "is" the same as the ordinary loop space functor. |
grouphom_compose {G H K : ooGroup}
(psi : ooGroupHom H K) (phi : ooGroupHom G H)
: ooGroupHom G K
:= pmap_compose psi phi. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | grouphom_compose | null |
group_loops_functor_compose
{X Y Z : pType}
(psi : Y ->* Z) (phi : X ->* Y)
: grouphom_compose (group_loops_functor psi) (group_loops_functor phi)
== group_loops_functor (pmap_compose psi phi). | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | group_loops_functor_compose | *** Functoriality |
grouphom_idmap (G : ooGroup) : ooGroupHom G G
:= pmap_idmap. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | grouphom_idmap | null |
group_loops_functor_idmap {X : pType}
: grouphom_idmap (group_loops X)
== group_loops_functor (Id (A:=pType) _). | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | group_loops_functor_idmap | null |
compose_grouphom {G H K : ooGroup}
(psi : ooGroupHom H K) (phi : ooGroupHom G H)
: grouphom_compose psi phi == psi o phi. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | compose_grouphom | null |
idmap_grouphom (G : ooGroup)
: grouphom_idmap G == idmap. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | idmap_grouphom | null |
grouphom_pp {G H} (phi : ooGroupHom G H) (g1 g2 : G)
: phi (g1 @ g2) = phi g1 @ phi g2. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | grouphom_pp | null |
grouphom_V {G H} (phi : ooGroupHom G H) (g : G)
: phi g^ = (phi g)^. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | grouphom_V | null |
grouphom_1 {G H} (phi : ooGroupHom G H)
: phi 1 = 1. | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | grouphom_1 | null |
grouphom_pp_p {G H} (phi : ooGroupHom G H) (g1 g2 g3 : G)
: grouphom_pp phi (g1 @ g2) g3
@ whiskerR (grouphom_pp phi g1 g2) (phi g3)
@ concat_pp_p (phi g1) (phi g2) (phi g3)
= ap phi (concat_pp_p g1 g2 g3)
@ grouphom_pp phi g1 (g2 @ g3)
@ whiskerL (phi g1) (grouphom_pp phi g2 g3). | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | grouphom_pp_p | null |
in_coset : G -> G -> Type
:= fun g1 g2 => hfiber incl (g1 @ g2^). | Definition | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | in_coset | A subgroup induces an equivalence relation on the ambient group, whose equivalence classes are called "cosets". |
#[export] Instance ishprop_in_coset : is_mere_relation G in_coset. | Instance | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | ishprop_in_coset | null |
#[export] Instance reflexive_coset : Reflexive in_coset. | Instance | Algebra | [
"Basics",
"Types",
"Pointed",
"Truncations",
"Homotopy",
"Algebra",
"WildCat"
] | theories/Algebra/ooGroup.v | reflexive_coset | null |
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