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Work on pointed naturality of smash-C

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This commit is contained in:
Floris van Doorn 2017-12-08 15:30:34 +01:00
parent 5bfb6e8d15
commit 743985e3d8
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32
Notes/smash.tex
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@ -573,7 +573,7 @@ We define the pointed equivalences:
\begin{thm}\label{thm:smash-functor-right}
Given pointed types $A$, $B$ and $C$, the functorial action of the smash product induces a map
$$({-})\smsh C:(A\pmap B)\pmap(A\smsh C\pmap B\smsh C)$$
which is natural in $A$, $B$ and dinatural in $C$.
which is natural in $A$, pointed natural in $B$ and dinatural in $C$.
\end{thm}
The naturality and dinaturality means that the following squares commute for $f : A' \to A$ $g:B\to B'$ and $h:C\to C'$.
\begin{center}
@ -614,8 +614,8 @@ fill the following square (after reducing out the applications of function exten
\end{tikzcd}
\end{center}
The naturality in $B$ is almost the same: for the underlying homotopy we need to show
$(g \o k)\smsh C = g\smsh C \o k\smsh C$. For the pointedness we need to fill the following
square, which follows from \autoref{lem:smash-coh}.
$i:(g \o k)\smsh C = g\smsh C \o k\smsh C$. For the pointedness we need to fill the following
square, which follows from the left pentagon in \autoref{lem:smash-coh}.
\begin{center}
\begin{tikzcd}
(g \o 0)\smsh C \arrow[r, equals]\arrow[dd,equals] &
@ -625,6 +625,14 @@ square, which follows from \autoref{lem:smash-coh}.
0
\end{tikzcd}
\end{center}
To show that this naturality is pointed, we need to show that if $g=0$ then this homotopy is the same as the concatenation of the following pointed homotopies:
$$q:({-})\smsh C \circ (A \to 0)\sim ({-})\smsh C \circ 0 \sim 0 \sim 0 \circ ({-})\smsh C\sim 0\smsh B \circ ({-})\smsh C.$$
To show that the underlying homotopies are the same, we need to show that $i(0,f,\idfunc[C],\idfunc[C])$ is equal to the following concatenation of pointed homotopies
$$q(f):(0\circ f)\smsh C\sim 0\smsh C \sim 0 \sim 0 \circ f\smsh C\sim 0\smsh B \circ f\smsh C,$$
which is the right pentagons in \autoref{lem:smash-coh}.
To show that these pointed homotopies respect the basepoint in the same way, we need to show that (TODO)
``$R\mathrel\square(0\smsh C \circ t)\cdot q_0=L$ where $L$ and $R$ are the left and right pentagons applied to $0$ and $\square$ is whiskering.''
The dinaturality in $C$ is a bit harder. For the underlying homotopy we need to show
$B\smsh h\o k\smsh C=k\smsh C'\o A\smsh h$. This follows by applying interchange twice:
$$B\smsh h\o k\smsh C\sim(\idfunc[B]\o k)\smsh(h\o\idfunc[C])\sim(k\o\idfunc[A])\smsh(\idfunc[C']\o h)\sim k\smsh C'\o A\smsh h.$$
@ -692,10 +700,9 @@ Note: $\eta$ is also dinatural in $B$, but we don't need this.
$\epsilon (x):C$ by induction on $x$. If $x\jdeq(f,b)$ we set $\epsilon(f,b)\defeq f(b)$. If $x$
is either $\auxl$ or $\auxr$ then we set $\epsilon (x)\defeq c_0:C$. If $x$ varies over $\gluel_f$
then we need to show that $f(b_0)=c_0$, which is true by $f_0$. If $x$ varies over $\gluer_b$ we
need to show that $0(b)=c_0$ which is true by reflexivity. The map $\epsilon$ defined as such is trivially a pointed map,
which defines the counit.
need to show that $0(b)=c_0$ which is true by reflexivity. Now $\epsilon_0\defeq 1:\epsilon(0_{B,C},b_0)=c_0$ shows that $\epsilon$ is pointed.
Now we need to show that the unit and counit are (di)natural. (TODO).
Now we need to show that the counit is natural in $B$ and pointed natural in $C$.
Finally, we need to show the unit-counit laws. For the underlying homotopy of the first one, let
$f:A\to B$. We need to show that $p_f:\epsilon\o\eta f\sim f$. We define $p_f(a)=1:\epsilon(f,a)=f(a)$. To show that $p_f$ is a pointed homotopy, we need to show that
@ -745,7 +752,7 @@ $$(A\pmap B\pmap C)\lpmap{({-})\smsh B}(A\smsh B\pmap (B\pmap C)\smsh B)\lpmap{A
% \end{align*}
\end{proof}
\begin{lem}\label{lem:e-natural}
The function $e$ is natural in $A$, $B$ and $C$.
The function $e$ is natural in $A$, $B$ and pointed natural in $C$.
\end{lem}
\begin{proof}
\textbf{Naturality of $e$ in $A$}. Suppose that $f:A'\pmap A$. Then the following diagram commutes. The left square commutes by naturality of $({-})\smsh B$ in the first argument and the right square commutes because composition on the left commutes with composition on the right.
@ -771,6 +778,7 @@ $$(A\pmap B\pmap C)\lpmap{({-})\smsh B}(A\smsh B\pmap (B\pmap C)\smsh B)\lpmap{A
(A\smsh B\pmap C')
\end{tikzcd}
\end{center}
Pointed naturality: TODO.
\textbf{Naturality of $e$ in $B$}. Suppose that $f:B'\pmap B$. Here the diagram is a bit more
complicated, since $({-})\smsh B$ is dinatural (instead of natural) in $B$. Then we get the
@ -796,15 +804,7 @@ argument, the back square commutes because composition on the left commutes with
inductive type to prove that.
\end{rmk}
\begin{lem}\label{lem:e-pointed-natural}
The function $e$ is pointed natural in $C$.
\end{lem}
\begin{proof}
(TODO)
\end{proof}
\section{Symmetric monoidality}
\section{Symmetric monoid product}
We aim to prove that the smash product is a (1-coherent) symmetric monoidal product [REF: Brunerie] for pointed types, i.e., that
\[(\type^*,\, \bool,\, \smsh,\, \alpha,\, \lambda,\, \rho,\, \gamma)\]
is a symmetric monoidal category, with the type of booleans $\bool$ (pointed in $0_\bool$) as unit, and for suitable instances of $\alpha$, $\lambda$, $\rho$ and $\gamma$ witnessing associativity, left- and right unitality and the braiding for $\smsh$ and satisfying appropriate coherence relations (associativity pentagon; unitors triangle; braiding-unitors triangle; associativity-braiding hexagon; double braiding).

15
homotopy/EM.hlean
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@ -655,8 +655,21 @@ namespace EM
definition cokernel {G H : AbGroup} (φ : G →g H) : AbGroup :=
quotient_ab_group (image φ)
/- todo: in algebra/quotient_group, do the first steps without assuming that N is normal,
then this is qg for (image φ) in H -/
definition image_cosets {G H : Group} (φ : G →g H) : Set* :=
sorry
definition homotopy_group_EMadd1_functor1 {G H : AbGroup} (φ : G →g H) (n : ℕ) :
πg[n+1] (pfiber (EMadd1_functor φ (n+1))) ≃g cokernel φ :=
sorry
definition homotopy_group_EMadd1_functor2 {G H : AbGroup} (φ : G →g H) (n : ℕ) :
πg[n+1] (pfiber (EMadd1_functor φ n)) ≃g Kernel φ :=
sorry
definition trunc_fiber_EM1_functor {G H : Group} (φ : G →g H) :
ptrunc 0 (pfiber (EM1_functor φ)) ≃* sorry :=
ptrunc 0 (pfiber (EM1_functor φ)) ≃* image_cosets φ :=
sorry
end EM

53
homotopy/smash.hlean
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@ -1,6 +1,6 @@
-- Authors: Floris van Doorn
import homotopy.smash types.pointed2 .pushout homotopy.red_susp
import homotopy.smash types.pointed2 .pushout homotopy.red_susp ..pointed
open bool pointed eq equiv is_equiv sum bool prod unit circle cofiber prod.ops wedge is_trunc
function red_susp unit
@ -536,14 +536,15 @@ namespace smash
square (p₁ ⬝ p₁⁻¹) p₂⁻¹ p₂ idp :=
proof square_of_eq (con.right_inv p₁ ⬝ (con.right_inv p₂)⁻¹) qed
private definition my_cube_fillerl {A B C : Type} {g : B → C} {f : A → B} {a₁ a₂ : A} {b₀ : B}
{p : f ~ λa, b₀} {q : Πa, g (f a) = g b₀} (r : (λa, ap g (p a)) ~ q) :
private definition my_cube_fillerl {B C : Type} {g : B → C} {fa₁ fa₂ b₀ : B}
{pa₁ : fa₁ = b₀} {pa₂ : fa₂ = b₀} {qa₁ : g (fa₁) = g b₀} {qa₂ : g (fa₂) = g b₀}
(ra₁ : ap g (pa₁) = qa₁) (ra₂ : ap g (pa₂) = qa₂) :
cube (hrfl ⬝hp (ra₁)⁻¹) hrfl
(my_squarel (qa₁) (qa₂)) (aps g (my_squarel (pa₁) (pa₂)))
(hrfl ⬝hp (!ap_con ⬝ whisker_left _ !ap_inv ⬝ (ra₁) ◾ (ra₂)⁻²)⁻¹)
(hrfl ⬝hp (ra₂)⁻²⁻¹ ⬝hp !ap_inv⁻¹) :=
begin
induction r using homotopy.rec_on_idp, induction p using homotopy.rec_on_idp_left, exact idc
induction ra₂, induction ra₁, induction pa₂, induction pa₁, exact idc
end
private definition my_cube_fillerr {B C : Type} {g : B → C} {b₀ bl br : B}
@ -615,11 +616,11 @@ namespace smash
refine !con.assoc ⬝ _, apply con_eq_of_eq_inv_con,
refine whisker_right _ _ ⬝ _, rotate 1, rexact functor_gluer'2_same f' g' (g b₀),
refine !inv_con_cancel_right ⬝ _,
exact sorry, -- TODO: FIX, the proof below should work (with some small changes)
-- refine _ ⬝ whisker_left _ _,
-- rotate 2, refine ap (whisker_left _) _, symmetry, exact !idp_con ⬝ !idp_con ⬝ !whisker_right_idp ⬝ !idp_con,
-- symmetry, apply whisker_left_idp
}
refine !whisker_left_idp⁻¹ ⬝ _,
refine !con_idp ⬝ _,
apply whisker_left,
apply ap (whisker_left idp),
exact (!idp_con ⬝ !idp_con ⬝ !whisker_right_idp ⬝ !idp_con)⁻¹ }
end
/- a version where the left maps are identities -/
@ -725,7 +726,7 @@ namespace smash
/- Using these lemmas we show that smash_functor_left is natural in all arguments -/
definition smash_functor_left_natural_left (f : A' →* A) :
definition smash_functor_left_natural_left (B C : Type*) (f : A' →* A) :
psquare (smash_functor_left A B C) (smash_functor_left A' B C)
(ppcompose_right f) (ppcompose_right (f ∧→ pid C)) :=
begin
@ -739,7 +740,7 @@ namespace smash
apply smash_functor_pcompose_pconst1_pid }
end
definition smash_functor_left_natural_middle (f : B →* B') :
definition smash_functor_left_natural_middle (A C : Type*) (f : B →* B') :
psquare (smash_functor_left A B C) (smash_functor_left A B' C)
(ppcompose_left f) (ppcompose_left (f ∧→ pid C)) :=
begin
@ -753,7 +754,7 @@ namespace smash
apply smash_functor_pcompose_pconst2_pid }
end
definition smash_functor_left_natural_right (f : C →* C') :
definition smash_functor_left_natural_right (A B : Type*) (f : C →* C') :
psquare (smash_functor_left A B C) (ppcompose_right (pid A ∧→ f))
(smash_functor_left A B C') (ppcompose_left (pid B ∧→ f)) :=
begin
@ -772,6 +773,18 @@ namespace smash
refine idp ◾** !refl_trans ⬝ !trans_left_inv }
end
definition smash_functor_left_natural_middle_phomotopy (A C : Type*) {f f' : B →* B'}
(p : f ~* f') : smash_functor_left_natural_middle A C f =
ppcompose_left_phomotopy p ⬝ph* smash_functor_left_natural_middle A C f' ⬝hp*
ppcompose_left_phomotopy (p ∧~ phomotopy.rfl) :=
begin
induction p using phomotopy_rec_idp,
symmetry,
refine !ppcompose_left_phomotopy_refl ◾ph* idp ◾hp*
(ap ppcompose_left_phomotopy !smash_functor_phomotopy_refl ⬝
!ppcompose_left_phomotopy_refl) ⬝ _,
exact !rfl_phomotopy_hconcat ⬝ !hconcat_phomotopy_rfl
end
/- the following is not really used, but a symmetric version of the natural equivalence
smash_functor_left -/
@ -877,11 +890,11 @@ namespace smash
refine !con.assoc ⬝ _, apply con_eq_of_eq_inv_con,
refine whisker_right _ _ ⬝ _, rotate 1, rexact functor_gluel'2_same f' g (f a₀),
refine !inv_con_cancel_right ⬝ _,
exact sorry, -- TODO: FIX, the proof below should work
-- refine _ ⬝ whisker_left _ _,
-- rotate 2, refine ap (whisker_left _) _, symmetry, exact !idp_con ⬝ !idp_con ⬝ !whisker_right_idp ⬝ !idp_con,
-- symmetry, apply whisker_left_idp
}
refine !whisker_left_idp⁻¹ ⬝ _,
refine !con_idp ⬝ _,
apply whisker_left,
apply ap (whisker_left idp),
exact (!idp_con ⬝ !idp_con ⬝ !whisker_right_idp ⬝ !idp_con)⁻¹ }
end
/- a version where the left maps are identities -/
@ -985,7 +998,7 @@ namespace smash
smash_functor_pcompose_pconst3 (pid A) (pid A) g
/- Using these lemmas we show that smash_functor_right is natural in all arguments -/
definition smash_functor_right_natural_right (f : C →* C') :
definition smash_functor_right_natural_right (A B : Type*) (f : C →* C') :
psquare (smash_functor_right A B C) (smash_functor_right A B C')
(ppcompose_left f) (ppcompose_left (pid A ∧→ f)) :=
begin
@ -999,7 +1012,7 @@ namespace smash
apply smash_functor_pcompose_pconst4_pid }
end
definition smash_functor_right_natural_middle (f : B' →* B) :
definition smash_functor_right_natural_middle (A C : Type*) (f : B' →* B) :
psquare (smash_functor_right A B C) (smash_functor_right A B' C)
(ppcompose_right f) (ppcompose_right (pid A ∧→ f)) :=
begin
@ -1013,7 +1026,7 @@ namespace smash
apply smash_functor_pcompose_pconst3_pid }
end
definition smash_functor_right_natural_left (f : A →* A') :
definition smash_functor_right_natural_left (B C : Type*) (f : A →* A') :
psquare (smash_functor_right A B C) (ppcompose_right (f ∧→ (pid B)))
(smash_functor_right A' B C) (ppcompose_left (f ∧→ (pid C))) :=
begin

66
homotopy/smash_adjoint.hlean
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@ -53,8 +53,10 @@ namespace smash
end
definition smash_pmap_unit_pt_natural [constructor] (B : Type*) (f : A →* A') :
smash_functor_pid_pinr' B f pt ⬝* pwhisker_left (smash_functor f (pid B)) (smash_pmap_unit_pt A B) ⬝*
pcompose_pconst (f ∧→ (pid B)) = pinr_phomotopy (respect_pt f) ⬝* smash_pmap_unit_pt A' B :=
smash_functor_pid_pinr' B f pt ⬝*
pwhisker_left (smash_functor f (pid B)) (smash_pmap_unit_pt A B) ⬝*
pcompose_pconst (f ∧→ (pid B)) =
pinr_phomotopy (respect_pt f) ⬝* smash_pmap_unit_pt A' B :=
begin
induction f with f f₀, induction A' with A' a₀', esimp at *,
induction f₀, refine _ ⬝ !refl_trans⁻¹,
@ -147,7 +149,7 @@ namespace smash
refine _ ⬝ (ap_is_constant respect_pt _)⁻¹, refine !idp_con⁻¹ }
end
definition smash_pmap_counit_natural_left (g : B →* B') :
definition smash_pmap_counit_natural_left (C : Type*) (g : B →* B') :
psquare (pid (ppmap B' C) ∧→ g) (smash_pmap_counit B C)
(ppcompose_right g ∧→ pid B) (smash_pmap_counit B' C) :=
begin
@ -336,20 +338,20 @@ namespace smash
definition smash_pelim_natural_left (B C : Type*) (f : A' →* A) :
psquare (smash_pelim A B C) (smash_pelim A' B C)
(ppcompose_right f) (ppcompose_right (f ∧→ pid B)) :=
smash_functor_left_natural_left f ⬝h* !ppcompose_left_ppcompose_right
smash_functor_left_natural_left (ppmap B C) B f ⬝h* !ppcompose_left_ppcompose_right
definition smash_pelim_natural_middle (A C : Type*) (f : B' →* B) :
psquare (smash_pelim A B C) (smash_pelim A B' C)
(ppcompose_left (ppcompose_right f)) (ppcompose_right (pid A ∧→ f)) :=
pwhisker_tl _ !ppcompose_left_ppcompose_right ⬝*
(!smash_functor_left_natural_right⁻¹* ⬝pv*
smash_functor_left_natural_middle (ppcompose_right f) ⬝h*
smash_functor_left_natural_middle _ _ (ppcompose_right f) ⬝h*
ppcompose_left_psquare !smash_pmap_counit_natural_left)
definition smash_pelim_natural_right (f : C →* C') :
definition smash_pelim_natural_right (A B : Type*) (f : C →* C') :
psquare (smash_pelim A B C) (smash_pelim A B C')
(ppcompose_left (ppcompose_left f)) (ppcompose_left f) :=
smash_functor_left_natural_middle (ppcompose_left f) ⬝h*
smash_functor_left_natural_middle _ _ (ppcompose_left f) ⬝h*
ppcompose_left_psquare (smash_pmap_counit_natural_right _ f)
definition smash_pelim_natural_lm (C : Type*) (f : A' →* A) (g : B' →* B) :
@ -393,28 +395,29 @@ namespace smash
(smash_adjoint_pmap A B C')⁻¹ᵉ* (ppcompose_left f ∘* g) :=
smash_elim_natural_right f g
definition smash_adjoint_pmap_inv_natural_right [constructor] {A B C C' : Type*} (f : C →* C') :
ppcompose_left f ∘* smash_adjoint_pmap_inv A B C ~*
smash_adjoint_pmap_inv A B C' ∘* ppcompose_left (ppcompose_left f) :=
smash_pelim_natural_right f
definition smash_adjoint_pmap_inv_natural_right [constructor] (A B : Type*) (f : C →* C') :
psquare (smash_adjoint_pmap_inv A B C) (smash_adjoint_pmap_inv A B C')
(ppcompose_left (ppcompose_left f)) (ppcompose_left f) :=
smash_pelim_natural_right A B f
definition smash_adjoint_pmap_natural_right [constructor] {A B C C' : Type*} (f : C →* C') :
ppcompose_left (ppcompose_left f) ∘* smash_adjoint_pmap A B C ~*
smash_adjoint_pmap A B C' ∘* ppcompose_left f :=
(smash_adjoint_pmap_inv_natural_right f)⁻¹ʰ*
definition smash_adjoint_pmap_natural_right [constructor] (A B : Type*) (f : C →* C') :
psquare (smash_adjoint_pmap A B C) (smash_adjoint_pmap A B C')
(ppcompose_left f) (ppcompose_left (ppcompose_left f)) :=
(smash_adjoint_pmap_inv_natural_right A B f)⁻¹ʰ*
definition smash_adjoint_pmap_natural_lm (C : Type*) (f : A →* A') (g : B →* B') :
psquare (smash_adjoint_pmap A' B' C) (smash_adjoint_pmap A B C)
(ppcompose_right (f ∧→ g)) (ppcompose_left (ppcompose_right g) ∘* ppcompose_right f) :=
(smash_pelim_natural_lm C f g)⁻¹ʰ*
/- some naturalities we skipped, but are now easier to prove -/
definition smash_elim_inv_natural_middle (f : B' →* B)
(g : A ∧ B →* C) : ppcompose_right f ∘* smash_elim_inv g ~* smash_elim_inv (g ∘* pid A ∧→ f) :=
!pcompose_pid⁻¹* ⬝* !passoc ⬝* phomotopy_of_eq (smash_adjoint_pmap_natural_lm C (pid A) f g)
definition smash_pmap_unit_natural_left (f : B →* B') :
ppcompose_left (pid A ∧→ f) ∘* smash_pmap_unit A B ~*
ppcompose_right f ∘* smash_pmap_unit A B' :=
psquare (smash_pmap_unit A B) (ppcompose_right f)
(smash_pmap_unit A B') (ppcompose_left (pid A ∧→ f)) :=
begin
refine pwhisker_left _ !smash_pelim_inv_pid⁻¹* ⬝* _ ⬝* pwhisker_left _ !smash_pelim_inv_pid,
refine !smash_elim_inv_natural_right ⬝* _ ⬝* !smash_elim_inv_natural_middle⁻¹*,
@ -461,7 +464,17 @@ namespace smash
definition smash_assoc_elim_natural_right_pt (f : X →* X') (g : A ∧ (B ∧ C) →* X) :
f ∘* smash_assoc_elim_pequiv A B C X g ~* smash_assoc_elim_pequiv A B C X' (f ∘* g) :=
phomotopy_of_eq (smash_assoc_elim_natural_right A B C f g)
begin
refine !smash_adjoint_pmap_inv_natural_right_pt ⬝* _,
apply smash_elim_phomotopy,
refine !smash_adjoint_pmap_inv_natural_right_pt ⬝* _,
apply smash_elim_phomotopy,
refine !passoc⁻¹* ⬝* _,
refine pwhisker_right _ !smash_adjoint_pmap_natural_right ⬝* _,
refine !passoc ⬝* _,
apply pwhisker_left,
refine !smash_adjoint_pmap_natural_right_pt
end
definition smash_assoc_elim_inv_natural_right_pt (f : X →* X') (g : (A ∧ B) ∧ C →* X) :
f ∘* (smash_assoc_elim_pequiv A B C X)⁻¹ᵉ* g ~*
@ -517,9 +530,9 @@ namespace smash
definition smash_assoc_elim_right_natural_right (A B C D : Type*) (f : X →* X') :
psquare (smash_assoc_elim_right_pequiv A B C D X) (smash_assoc_elim_right_pequiv A B C D X')
(ppcompose_left f) (ppcompose_left f) :=
smash_adjoint_pmap_natural_right f ⬝h*
smash_adjoint_pmap_natural_right (A ∧ (B ∧ C)) D f ⬝h*
smash_assoc_elim_natural_right A B C (ppcompose_left f) ⬝h*
smash_adjoint_pmap_inv_natural_right f
smash_adjoint_pmap_inv_natural_right ((A ∧ B) ∧ C) D f
definition smash_assoc_smash_functor (A B C D : Type*) :
smash_assoc A B C ∧→ pid D ~* !smash_assoc_elim_right_pequiv (pid _) :=
@ -534,7 +547,7 @@ namespace smash
definition ppcompose_right_smash_assoc (A B C X : Type*) :
ppcompose_right (smash_assoc A B C) ~* smash_assoc_elim_pequiv A B C X :=
sorry -- one hole left
sorry
definition smash_functor_smash_assoc (A B C D : Type*) :
pid A ∧→ smash_assoc B C D ~* !smash_assoc_elim_left_pequiv (pid _) :=
@ -543,7 +556,7 @@ namespace smash
refine pap (!smash_adjoint_pmap_inv ∘* ppcompose_left _) !smash_pelim_inv_pid ⬝* _,
refine pap !smash_adjoint_pmap_inv (pwhisker_right _ !ppcompose_right_smash_assoc⁻¹* ⬝*
!smash_pmap_unit_natural_left⁻¹*) ⬝* _,
refine phomotopy_of_eq (smash_adjoint_pmap_inv_natural_right (pid A ∧→ smash_assoc B C D)
refine phomotopy_of_eq (smash_adjoint_pmap_inv_natural_right _ _ (pid A ∧→ smash_assoc B C D)
!smash_pmap_unit)⁻¹ ⬝* _,
refine pwhisker_left _ _ ⬝* !pcompose_pid,
apply smash_pmap_unit_counit
@ -589,10 +602,10 @@ namespace smash
definition smash_susp_elim_natural_right (A B : Type*) (f : X →* X') :
psquare (smash_susp_elim_pequiv A B X) (smash_susp_elim_pequiv A B X')
(ppcompose_left f) (ppcompose_left f) :=
smash_adjoint_pmap_natural_right f ⬝h*
smash_adjoint_pmap_natural_right (⅀ A) B f ⬝h*
susp_adjoint_loop_natural_right (ppcompose_left f) ⬝h*
ppcompose_left_psquare (loop_pmap_commute_natural_right B f) ⬝h*
(smash_adjoint_pmap_natural_right (Ω→ f))⁻¹ʰ* ⬝h*
(smash_adjoint_pmap_natural_right A B (Ω→ f))⁻¹ʰ* ⬝h*
(susp_adjoint_loop_natural_right f)⁻¹ʰ*
definition smash_susp_elim_natural_left (X : Type*) (f : A' →* A) (g : B' →* B) :
@ -640,6 +653,11 @@ namespace smash
... ≃* ⅀(B ∧ A) : susp_smash B A
... ≃* ⅀(A ∧ B) : susp_pequiv (smash_comm B A)
definition smash_susp_natural (f : A →* A') (g : B →* B') :
psquare (smash_susp A B) (smash_susp A' B') (f ∧→ ⅀→g) (⅀→ (f ∧→ g)) :=
sorry
definition susp_smash_move (A B : Type*) : ⅀ A ∧ B ≃* A ∧ ⅀ B :=
susp_smash A B ⬝e* (smash_susp A B)⁻¹ᵉ*

9
move_to_lib.hlean
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@ -932,6 +932,9 @@ namespace is_conn
definition is_conn_zero_pointed {A : Type*} (p : Πa a' : A, ∥ a = a' ∥) : is_conn 0 A :=
is_conn_zero (tr pt) p
definition is_conn_zero_pointed' {A : Type*} (p : Πa : A, ∥ a = pt ∥) : is_conn 0 A :=
is_conn_zero_pointed (λa a', tconcat (p a) (tinverse (p a')))
definition is_conn_fiber (n : ℕ₋₂) {A B : Type} (f : A → B) (b : B) [is_conn n A] [is_conn (n.+1) B] :
is_conn n (fiber f b) :=
is_conn_equiv_closed_rev _ !fiber.sigma_char _
@ -951,10 +954,8 @@ namespace misc
pType.mk (Σ(a : A), merely (pt = a)) ⟨pt, tr idp⟩
lemma is_conn_component [instance] (A : Type*) : is_conn 0 (component A) :=
is_contr.mk (tr pt)
begin
intro x, induction x with x, induction x with a p, induction p with p, induction p, reflexivity
end
is_conn_zero_pointed'
begin intro x, induction x with a p, induction p with p, induction p, exact tidp end
definition component_incl [constructor] (A : Type*) : component A →* A :=
pmap.mk pr1 idp

305
pointed.hlean
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@ -8,6 +8,43 @@ open pointed eq equiv function is_equiv unit is_trunc trunc nat algebra sigma gr
namespace pointed
definition phomotopy_mk_eq {A B : Type*} {f g : A →* B} {h k : f ~ g}
{h₀ : h pt ⬝ respect_pt g = respect_pt f} {k₀ : k pt ⬝ respect_pt g = respect_pt f} (p : h ~ k)
(q : whisker_right (respect_pt g) (p pt) ⬝ k₀ = h₀) : phomotopy.mk h h₀ = phomotopy.mk k k₀ :=
phomotopy_eq p (idp ◾ to_right_inv !eq_con_inv_equiv_con_eq _ ⬝
q ⬝ (to_right_inv !eq_con_inv_equiv_con_eq _)⁻¹)
section phsquare
/-
Squares of pointed homotopies
-/
variables {A : Type*} {P : A → Type} {p₀ : P pt}
{f f' f₀₀ f₂₀ f₄₀ f₀₂ f₂₂ f₄₂ f₀₄ f₂₄ f₄₄ : ppi P p₀}
{p₁₀ : f₀₀ ~* f₂₀} {p₃₀ : f₂₀ ~* f₄₀}
{p₀₁ : f₀₀ ~* f₀₂} {p₂₁ : f₂₀ ~* f₂₂} {p₄₁ : f₄₀ ~* f₄₂}
{p₁₂ : f₀₂ ~* f₂₂} {p₃₂ : f₂₂ ~* f₄₂}
{p₀₃ : f₀₂ ~* f₀₄} {p₂₃ : f₂₂ ~* f₂₄} {p₄₃ : f₄₂ ~* f₄₄}
{p₁₄ : f₀₄ ~* f₂₄} {p₃₄ : f₂₄ ~* f₄₄}
definition phtranspose (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) : phsquare p₀₁ p₂₁ p₁₀ p₁₂ :=
p⁻¹
definition eq_top_of_phsquare (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) : p₁₀ = p₀₁ ⬝* p₁₂ ⬝* p₂₁⁻¹* :=
eq_trans_symm_of_trans_eq p
definition eq_bot_of_phsquare (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) : p₁₂ = p₀₁⁻¹* ⬝* p₁₀ ⬝* p₂₁ :=
eq_symm_trans_of_trans_eq p⁻¹ ⬝ !trans_assoc⁻¹
definition eq_left_of_phsquare (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) : p₀₁ = p₁₀ ⬝* p₂₁ ⬝* p₁₂⁻¹* :=
eq_top_of_phsquare (phtranspose p)
definition eq_right_of_phsquare (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) : p₂₁ = p₁₀⁻¹* ⬝* p₀₁ ⬝* p₁₂ :=
eq_bot_of_phsquare (phtranspose p)
end phsquare
-- /- the pointed type of (unpointed) dependent maps -/
-- definition pupi [constructor] {A : Type} (P : A → Type*) : Type* :=
-- pointed.mk' (Πa, P a)
@ -33,12 +70,32 @@ namespace pointed
-- apply equiv_eq_closed_right, exact !idp_con⁻¹ }
-- end
/- todo: make type argument explicit in ppcompose_left and ppcompose_left_* -/
/- todo: delete papply_pcompose -/
/- todo: pmap_pbool_equiv is a special case of ppmap_pbool_pequiv. -/
definition ppcompose_left_pid [constructor] (A B : Type*) : ppcompose_left (pid B) ~* pid (ppmap A B) :=
phomotopy_mk_ppmap (λf, pid_pcompose f) (!trans_refl ⬝ !phomotopy_of_eq_of_phomotopy⁻¹)
definition ppcompose_right_pid [constructor] (A B : Type*) : ppcompose_right (pid A) ~* pid (ppmap A B) :=
phomotopy_mk_ppmap (λf, pcompose_pid f) (!trans_refl ⬝ !phomotopy_of_eq_of_phomotopy⁻¹)
section
variables {A A' : Type*} {P : A → Type} {P' : A' → Type} {p₀ : P pt} {p₀' : P' pt} {k l : ppi P p₀}
definition phomotopy_of_eq_inv (p : k = l) :
phomotopy_of_eq p⁻¹ = (phomotopy_of_eq p)⁻¹* :=
begin induction p, exact !refl_symm⁻¹ end
/- todo: replace regular pap -/
definition pap' (f : ppi P p₀ → ppi P' p₀') (p : k ~* l) : f k ~* f l :=
by induction p using phomotopy_rec_idp; reflexivity
definition phomotopy_of_eq_ap (f : ppi P p₀ → ppi P' p₀') (p : k = l) :
phomotopy_of_eq (ap f p) = pap' f (phomotopy_of_eq p) :=
begin induction p, exact !phomotopy_rec_idp_refl⁻¹ end
end
/- remove some duplicates: loop_ppmap_commute, loop_ppmap_pequiv, loop_ppmap_pequiv', pfunext -/
definition pfunext (X Y : Type*) : ppmap X (Ω Y) ≃* Ω (ppmap X Y) :=
(loop_ppmap_commute X Y)⁻¹ᵉ*
@ -147,13 +204,69 @@ namespace pointed
/- Homotopy between a function and its eta expansion -/
definition pmap_eta {X Y : Type*} (f : X →* Y) : f ~* pmap.mk f (respect_pt f) :=
definition pmap_eta [constructor] {X Y : Type*} (f : X →* Y) : f ~* pmap.mk f (respect_pt f) :=
begin
fapply phomotopy.mk,
reflexivity,
esimp, exact !idp_con
end
definition papply_point [constructor] (A B : Type*) : papply B pt ~* pconst (ppmap A B) B :=
phomotopy.mk (λf, respect_pt f) idp
definition pmap_swap_map [constructor] {A B C : Type*} (f : A →* ppmap B C) :
ppmap B (ppmap A C) :=
begin
fapply pmap.mk,
{ intro b, exact papply C b ∘* f },
{ apply eq_of_phomotopy, exact pwhisker_right f (papply_point B C) ⬝* !pconst_pcompose }
end
definition pmap_swap_map_pconst (A B C : Type*) :
pmap_swap_map (pconst A (ppmap B C)) ~* pconst B (ppmap A C) :=
begin
fapply phomotopy_mk_ppmap,
{ intro b, reflexivity },
{ refine !refl_trans ⬝ !phomotopy_of_eq_of_phomotopy⁻¹ }
end
definition papply_pmap_swap_map [constructor] {A B C : Type*} (f : A →* ppmap B C) (a : A) :
papply C a ∘* pmap_swap_map f ~* f a :=
begin
fapply phomotopy.mk,
{ intro b, reflexivity },
{ exact !idp_con ⬝ !ap_eq_of_phomotopy⁻¹ }
end
definition pmap_swap_map_pmap_swap_map {A B C : Type*} (f : A →* ppmap B C) :
pmap_swap_map (pmap_swap_map f) ~* f :=
begin
fapply phomotopy_mk_ppmap,
{ exact papply_pmap_swap_map f },
{ refine _ ⬝ !phomotopy_of_eq_of_phomotopy⁻¹,
fapply phomotopy_mk_eq, intro b, exact !idp_con,
refine !whisker_right_idp ◾ (!idp_con ◾ idp) ⬝ _ ⬝ !idp_con⁻¹ ◾ idp,
symmetry, exact sorry }
end
definition pmap_swap [constructor] (A B C : Type*) : ppmap A (ppmap B C) →* ppmap B (ppmap A C) :=
begin
fapply pmap.mk,
{ exact pmap_swap_map },
{ exact eq_of_phomotopy (pmap_swap_map_pconst A B C) }
end
definition pmap_swap_pequiv [constructor] (A B C : Type*) :
ppmap A (ppmap B C) ≃* ppmap B (ppmap A C) :=
begin
fapply pequiv_of_pmap,
{ exact pmap_swap A B C },
fapply adjointify,
{ exact pmap_swap B A C },
{ intro f, apply eq_of_phomotopy, exact pmap_swap_map_pmap_swap_map f },
{ intro f, apply eq_of_phomotopy, exact pmap_swap_map_pmap_swap_map f }
end
-- this should replace pnatural_square
definition pnatural_square2 {A B : Type} (X : B → Type*) (Y : B → Type*) {f g : A → B}
(h : Πa, X (f a) →* Y (g a)) {a a' : A} (p : a = a') :
@ -176,10 +289,39 @@ namespace pointed
section psquare
variables {A A' A₀₀ A₂₀ A₄₀ A₀₂ A₂₂ A₄₂ A₀₄ A₂₄ A₄₄ : Type*}
{f₁₀ f₁₀' : A₀₀ →* A₂₀} {f₃₀ : A₂₀ →* A₄₀}
{f₀₁ f₀₁' : A₀₀ →* A₀₂} {f₂₁ f₂₁' : A₂₀ →* A₂₂} {f₄₁ : A₄₀ →* A₄₂}
{f₀₁ f₀₁' : A₀₀ →* A₀₂} {f₂₁ f₂₁' : A₂₀ →* A₂₂} {f₄₁ f₄₁' : A₄₀ →* A₄₂}
{f₁₂ f₁₂' : A₀₂ →* A₂₂} {f₃₂ : A₂₂ →* A₄₂}
{f₀₃ : A₀₂ →* A₀₄} {f₂₃ : A₂₂ →* A₂₄} {f₄₃ : A₄₂ →* A₄₄}
{f₁₄ : A₀₄ →* A₂₄} {f₃₄ : A₂₄ →* A₄₄}
{f₁₄ f₁₄' : A₀₄ →* A₂₄} {f₃₄ : A₂₄ →* A₄₄}
definition phconst_square (f₀₁ : A₀₀ →* A₀₂) (f₂₁ : A₂₀ →* A₂₂) :
psquare (pconst A₀₀ A₂₀) (pconst A₀₂ A₂₂) f₀₁ f₂₁ :=
!pcompose_pconst ⬝* !pconst_pcompose⁻¹*
definition pvconst_square (f₁₀ : A₀₀ →* A₂₀) (f₁₂ : A₀₂ →* A₂₂) :
psquare f₁₀ f₁₂ (pconst A₀₀ A₀₂) (pconst A₂₀ A₂₂) :=
!pconst_pcompose ⬝* !pcompose_pconst⁻¹*
definition pvconst_square_pcompose (f₃₀ : A₂₀ →* A₄₀) (f₁₀ : A₀₀ →* A₂₀)
(f₃₂ : A₂₂ →* A₄₂) (f₁₂ : A₀₂ →* A₂₂) :
pvconst_square (f₃₀ ∘* f₁₀) (f₃₂ ∘* f₁₂) = pvconst_square f₁₀ f₁₂ ⬝h* pvconst_square f₃₀ f₃₂ :=
begin
refine eq_right_of_phsquare !passoc_pconst_left ◾** !passoc_pconst_right⁻¹⁻²** ⬝ idp ◾**
(!trans_symm ⬝ !trans_symm ◾** idp) ⬝ !trans_assoc⁻¹ ⬝ _ ◾** idp,
refine !trans_assoc⁻¹ ⬝ _ ◾** !pwhisker_left_symm⁻¹ ⬝ !trans_assoc ⬝
idp ◾** !pwhisker_left_trans⁻¹,
apply trans_symm_eq_of_eq_trans,
refine _ ⬝ idp ◾** !passoc_pconst_middle⁻¹ ⬝ !trans_assoc⁻¹ ⬝ !trans_assoc⁻¹,
refine _ ◾** idp ⬝ !trans_assoc,
refine idp ◾** _ ⬝ !trans_assoc⁻¹,
refine ap (pwhisker_right f₁₀) _ ⬝ !pwhisker_right_trans,
refine !trans_refl⁻¹ ⬝ idp ◾** !trans_left_inv⁻¹ ⬝ !trans_assoc⁻¹,
end
definition phconst_square_pcompose (f₀₃ : A₀₂ →* A₀₄) (f₁₀ : A₀₀ →* A₂₀)
(f₂₃ : A₂₂ →* A₂₄) (f₂₁ : A₂₀ →* A₂₂) :
phconst_square (f₀₃ ∘* f₀₁) (f₂₃ ∘* f₁₂) = phconst_square f₀₁ f₁₂ ⬝v* phconst_square f₀₃ f₂₃ :=
sorry
definition ptranspose (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : psquare f₀₁ f₂₁ f₁₀ f₁₂ :=
p⁻¹*
@ -187,9 +329,94 @@ namespace pointed
definition hsquare_of_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : hsquare f₁₀ f₁₂ f₀₁ f₂₁ :=
p
definition homotopy_group_functor_hsquare (n : ℕ) (h : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
psquare (π→[n] f₁₀) (π→[n] f₁₂)
(π→[n] f₀₁) (π→[n] f₂₁) :=
definition rfl_phomotopy_hconcat (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : phomotopy.rfl ⬝ph* p = p :=
idp ◾** (ap (pwhisker_left f₁₂) !refl_symm ⬝ !pwhisker_left_refl) ⬝ !trans_refl
definition hconcat_phomotopy_rfl (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : p ⬝hp* phomotopy.rfl = p :=
!pwhisker_right_refl ◾** idp ⬝ !refl_trans
definition rfl_phomotopy_vconcat (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : phomotopy.rfl ⬝pv* p = p :=
!pwhisker_left_refl ◾** idp ⬝ !refl_trans
definition vconcat_phomotopy_rfl (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : p ⬝vp* phomotopy.rfl = p :=
idp ◾** (ap (pwhisker_right f₀₁) !refl_symm ⬝ !pwhisker_right_refl) ⬝ !trans_refl
definition phomotopy_hconcat_phconcat (p : f₀₁' ~* f₀₁) (q : psquare f₁₀ f₁₂ f₀₁ f₂₁)
(r : psquare f₃₀ f₃₂ f₂₁ f₄₁) : (p ⬝ph* q) ⬝h* r = p ⬝ph* (q ⬝h* r) :=
begin
induction p using phomotopy_rec_idp,
exact !refl_phomotopy_hconcat ◾h* idp ⬝ !refl_phomotopy_hconcat⁻¹
end
definition phconcat_hconcat_phomotopy (p : psquare f₁₀ f₁₂ f₀₁ f₂₁)
(q : psquare f₃₀ f₃₂ f₂₁ f₄₁) (r : f₄₁' ~* f₄₁) : (p ⬝h* q) ⬝hp* r = p ⬝h* (q ⬝hp* r) :=
begin
induction r using phomotopy_rec_idp,
exact !hconcat_phomotopy_rfl ⬝ idp ◾h* !hconcat_phomotopy_rfl⁻¹
end
definition phomotopy_hconcat_phomotopy (p : f₀₁' ~* f₀₁) (q : psquare f₁₀ f₁₂ f₀₁ f₂₁)
(r : f₂₁' ~* f₂₁) : (p ⬝ph* q) ⬝hp* r = p ⬝ph* (q ⬝hp* r) :=
begin
induction r using phomotopy_rec_idp,
exact !hconcat_phomotopy_rfl ⬝ idp ◾ph* !hconcat_phomotopy_rfl⁻¹
end
definition hconcat_phomotopy_phconcat (p : psquare f₁₀ f₁₂ f₀₁ f₂₁)
(q : f₂₁' ~* f₂₁) (r : psquare f₃₀ f₃₂ f₂₁' f₄₁) : (p ⬝hp* q) ⬝h* r = p ⬝h* (q⁻¹* ⬝ph* r) :=
begin
induction q using phomotopy_rec_idp,
exact !hconcat_phomotopy_rfl ◾h* idp ⬝ idp ◾h* (!refl_symm ◾ph* idp ⬝ !refl_phomotopy_hconcat)⁻¹
end
definition phconcat_phomotopy_hconcat (p : psquare f₁₀ f₁₂ f₀₁ f₂₁)
(q : f₂₁ ~* f₂₁') (r : psquare f₃₀ f₃₂ f₂₁' f₄₁) : p ⬝h* (q ⬝ph* r) = (p ⬝hp* q⁻¹*) ⬝h* r :=
begin
induction q using phomotopy_rec_idp,
exact idp ◾h* !refl_phomotopy_hconcat ⬝ (idp ◾hp* !refl_symm ⬝ !hconcat_phomotopy_rfl)⁻¹ ◾h* idp
end
definition hconcat_phomotopy_hconcat_cancel (p : psquare f₁₀ f₁₂ f₀₁ f₂₁)
(q : f₂₁' ~* f₂₁) (r : psquare f₃₀ f₃₂ f₂₁ f₄₁) : (p ⬝hp* q) ⬝h* (q ⬝ph* r) = p ⬝h* r :=
begin
induction q using phomotopy_rec_idp,
exact !hconcat_phomotopy_rfl ◾h* !refl_phomotopy_hconcat
end
definition phomotopy_hconcat_phinverse {f₁₀ : A₀₀ ≃* A₂₀} {f₁₂ : A₀₂ ≃* A₂₂}
(p : f₀₁' ~* f₀₁) (q : psquare f₁₀ f₁₂ f₀₁ f₂₁) : (p ⬝ph* q)⁻¹ʰ* = q⁻¹ʰ* ⬝hp* p :=
begin
induction p using phomotopy_rec_idp,
exact !refl_phomotopy_hconcat⁻²ʰ* ⬝ !hconcat_phomotopy_rfl⁻¹
end
definition hconcat_phomotopy_phinverse {f₁₀ : A₀₀ ≃* A₂₀} {f₁₂ : A₀₂ ≃* A₂₂}
(p : psquare f₁₀ f₁₂ f₀₁ f₂₁) (q : f₂₁' ~* f₂₁) : (p ⬝hp* q)⁻¹ʰ* = q ⬝ph* p⁻¹ʰ* :=
begin
induction q using phomotopy_rec_idp,
exact !hconcat_phomotopy_rfl⁻²ʰ* ⬝ !refl_phomotopy_hconcat⁻¹
end
definition pvconst_square_phinverse (f₁₀ : A₀₀ ≃* A₂₀) (f₁₂ : A₀₂ ≃* A₂₂) :
(pvconst_square f₁₀ f₁₂)⁻¹ʰ* = pvconst_square f₁₀⁻¹ᵉ* f₁₂⁻¹ᵉ* :=
begin
exact sorry
end
definition ppcompose_left_phomotopy_hconcat (A : Type*) (p : f₀₁' ~* f₀₁)
(q : psquare f₁₀ f₁₂ f₀₁ f₂₁) : ppcompose_left_psquare (p ⬝ph* q) =
@ppcompose_left_phomotopy A _ _ _ _ p ⬝ph* ppcompose_left_psquare q :=
sorry --used
definition ppcompose_left_hconcat_phomotopy (A : Type*) (p : psquare f₁₀ f₁₂ f₀₁ f₂₁)
(q : f₂₁' ~* f₂₁) : ppcompose_left_psquare (p ⬝hp* q) =
ppcompose_left_psquare p ⬝hp* @ppcompose_left_phomotopy A _ _ _ _ q :=
sorry
definition ppcompose_left_pvconst_square (A : Type*) (f₁₀ : A₀₀ →* A₂₀) (f₁₂ : A₀₂ →* A₂₂) :
ppcompose_left_psquare (pvconst_square f₁₀ f₁₂) =
!ppcompose_left_pconst ⬝ph* pvconst_square (ppcompose_left f₁₀) (@ppcompose_left A _ _ f₁₂) ⬝hp*
!ppcompose_left_pconst :=
sorry
end psquare
@ -282,6 +509,72 @@ namespace pointed
A →* pointed.MK B (f pt) :=
pmap.mk f idp
definition papply_natural_right [constructor] {A B B' : Type*} (f : B →* B') (a : A) :
psquare (papply B a) (papply B' a) (ppcompose_left f) f :=
begin
fapply phomotopy.mk,
{ intro g, reflexivity },
{ refine !idp_con ⬝ !ap_eq_of_phomotopy ⬝ !idp_con⁻¹ }
end
-- definition foo {A B : Type} {f : A → B} {a₀ a₁ a₂ : A} {b₀ b₁ b₂ : B}
-- {p₀ : a₀ = a₀} {p₁ : a₀ = a₀} (q₀ : ap f p₀ = idp) (q₁ : ap f p₁ = idp) :
-- whisker_right (ap f p₁) (idp_con (ap f p₀⁻¹) ⬝ !ap_inv ⬝ q₀⁻²)⁻¹ ⬝ !con.assoc ⬝
-- whisker_left idp (con_eq_of_eq_con_inv (eq_con_inv_of_con_eq _)) ⬝ _
-- = !idp_con ⬝ q₁ :=
-- _
/-
whisker_right
(ap (λ f, pppi.to_fun f a)
(eq_of_phomotopy (pcompose_pconst (pconst B B'))))
(idp_con
(ap (λ y, pppi.to_fun y a)
(eq_of_phomotopy
(pconst_pcompose (ppi_const (λ a, B))))⁻¹) ⬝ ap_inv
(λ y, pppi.to_fun y a)
(eq_of_phomotopy
(pconst_pcompose (ppi_const (λ a, B)))) ⬝ inverse2
(ap_eq_of_phomotopy (pconst_pcompose (ppi_const (λ a, B)))
a))⁻¹ ⬝ (con.assoc
(ppi.to_fun (pvconst_square (papply B a) (papply B' a))
(ppi_const (λ a, B)))
(ap (λ f, pppi.to_fun f a)
(eq_of_phomotopy (pconst_pcompose (ppi_const (λ a, B))))⁻¹)
(ap (λ f, pppi.to_fun f a)
(eq_of_phomotopy
(pcompose_pconst (pconst B B')))) ⬝ whisker_left
(ppi.to_fun (pvconst_square (papply B a) (papply B' a))
(ppi_const (λ a, B)))
(con_eq_of_eq_con_inv
(eq_con_inv_of_con_eq
(pwhisker_left_1 B' (ppmap A B) (ppmap A B') (papply B' a)
(pconst (ppmap A B) (ppmap A B'))
(ppcompose_left (pconst B B'))
(ppcompose_left_pconst A B B')⁻¹*))) ⬝ respect_pt
(pvconst_square (papply B a) (papply B' a))) = idp_con
(ap (λ f, pppi.to_fun f a)
(eq_of_phomotopy
(pcompose_pconst (pconst B B')))) ⬝ ap_eq_of_phomotopy
(pcompose_pconst (pconst B B'))
a
-/
definition papply_natural_right_pconst {A : Type*} (B B' : Type*) (a : A) :
papply_natural_right (pconst B B') a = !ppcompose_left_pconst ⬝ph* !pvconst_square :=
begin
fapply phomotopy_mk_eq,
{ intro g, symmetry, refine !idp_con ⬝ !ap_inv ⬝ !ap_eq_of_phomotopy⁻² },
{
esimp [pvconst_square],
--esimp [inv_con_eq_of_eq_con, pwhisker_left_1]
exact sorry }
end
/- TODO: computation rule -/
open pi
definition fiberwise_pointed_map_rec {A : Type} {B : A → Type*}

2
spectrum/basic.hlean
View file

@ -749,7 +749,7 @@ namespace spectrum
begin
intro k,
note sq1 := homotopy_group_homomorphism_psquare (k+2) (ptranspose (smap.glue_square f (-n - 2 +[ℤ] k))),
note sq2 := homotopy_group_functor_hsquare (k+2) (ap1_psquare (ptransport_natural E F f (add.assoc (-n - 2) k 1))),
note sq2 := homotopy_group_functor_psquare (k+2) (ap1_psquare (ptransport_natural E F f (add.assoc (-n - 2) k 1))),
note sq3 := (homotopy_group_succ_in_natural (k+2) (f (-n - 2 +[ℤ] (k+1))))⁻¹ʰᵗʸʰ,
note sq4 := hsquare_of_psquare sq2,
note rect := sq1 ⬝htyh sq4 ⬝htyh sq3,

13
spectrum/smash.hlean
View file

@ -14,7 +14,8 @@ prespectrum.mk (λ z, X ∧ Y z) begin
exact !glue
end
definition smash_prespectrum_fun {X X' : Type*} {Y Y' : prespectrum} (f : X →* X') (g : Y →ₛ Y') : smash_prespectrum X Y →ₛ smash_prespectrum X' Y' :=
definition smash_prespectrum_fun {X X' : Type*} {Y Y' : prespectrum} (f : X →* X') (g : Y →ₛ Y') :
smash_prespectrum X Y →ₛ smash_prespectrum X' Y' :=
smap.mk (λn, smash_functor f (g n)) begin
intro n,
refine susp_to_loop_psquare _ _ _ _ _,
@ -22,12 +23,12 @@ smap.mk (λn, smash_functor f (g n)) begin
refine vconcat_phomotopy _ (smash_functor_split f (g (S n))),
refine phomotopy_vconcat (smash_functor_split f (susp_functor (g n))) _,
refine phconcat _ _,
let glue_adjoint := susp_pelim (Y n) (Y (S n)) (glue Y n),
exact pid X' ∧→ glue_adjoint,
exact smash_functor_psquare (pvrefl f) (phrefl glue_adjoint),
refine smash_functor_psquare (phrefl (pid X')) _,
let glue_adjoint := susp_pelim (Y' n) (Y' (S n)) (glue Y' n),
exact pid X ∧→ glue_adjoint,
refine smash_functor_psquare (phrefl (pid X)) _,
refine loop_to_susp_square _ _ _ _ _,
exact smap.glue_square g n
exact smap.glue_square g n,
exact smash_functor_psquare (pvrefl f) (phrefl glue_adjoint)
end
/- smash of a spectrum and a type -/