From 39883fd3ee03369f8e09efc6183cc6729f1e1312 Mon Sep 17 00:00:00 2001 From: Yuri Sulyma Date: Fri, 9 Jun 2017 12:24:33 -0600 Subject: [PATCH] A bit of code for incoherent homotopies between maps of spectra --- homotopy/spectrum.hlean | 20 ++++++++++++++++++++ 1 file changed, 20 insertions(+) diff --git a/homotopy/spectrum.hlean b/homotopy/spectrum.hlean index 0be9bea..c18227d 100644 --- a/homotopy/spectrum.hlean +++ b/homotopy/spectrum.hlean @@ -214,6 +214,19 @@ namespace spectrum exact phhinverse ((shomotopy.glue_homotopy p) n) end + -- incoherent homotopies. this is a bit gross, but + -- a) we don't need the higher coherences for most basic things + -- (you need it for higher algebra, e.g. power operations) + -- b) homotopies of maps between spectra are really hard + + structure shomotopy_incoh {N : succ_str} {E F : gen_prespectrum N} (f g : E →ₛ F) := + (to_phomotopy : Πn, f n ~* g n) + + infix ` ~ₛi `:50 := shomotopy_incoh + + definition shomotopy_to_incoh [coercion] {N : succ_str} {E F : gen_prespectrum N} {f g : E →ₛ F} (p : f ~ₛ g) : shomotopy_incoh f g := + shomotopy_incoh.mk (λn, (shomotopy.to_phomotopy p) n) + ------------------------------ -- Suspension prespectra ------------------------------ @@ -255,6 +268,13 @@ namespace spectrum notation `πₛ→[`:95 n:0 `]`:0 := shomotopy_group_fun n + -- what an awful name + definition shomotopy_group_fun_shomotopy_incoh {E F : spectrum} {f g : E →ₛ F} (n : ℤ) (p : f ~ₛi g) : πₛ→[n] f ~ πₛ→[n] g := + begin + refine homotopy_group_functor_phomotopy 2 _, + exact (shomotopy_incoh.to_phomotopy p) (2 - n) + end + /- homotopy group of a prespectrum -/ definition pshomotopy_group (n : ℤ) (E : prespectrum) : AbGroup :=