42 lines
1.6 KiB
Text
42 lines
1.6 KiB
Text
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import .spectrification .smash_adjoint
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open pointed is_equiv equiv eq susp succ_str smash int
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namespace spectrum
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/- Smash product of a prespectrum and a type -/
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definition smash_prespectrum (X : Type*) (Y : prespectrum) : prespectrum :=
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prespectrum.mk (λ z, X ∧ Y z) begin
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intro n, refine loop_psusp_pintro (X ∧ Y n) (X ∧ Y (n + 1)) _,
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refine _ ∘* (smash_psusp X (Y n))⁻¹ᵉ*,
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refine smash_functor !pid _,
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refine psusp_pelim (Y n) (Y (n + 1)) _,
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exact !glue
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end
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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' :=
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smap.mk (λn, smash_functor f (g n)) begin
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intro n,
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refine susp_to_loop_psquare _ _ _ _ _,
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refine pvconcat (psquare_transpose (phinverse (smash_psusp_natural f (g n)))) _,
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refine vconcat_phomotopy _ (smash_functor_split f (g (S n))),
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refine phomotopy_vconcat (smash_functor_split f (psusp_functor (g n))) _,
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refine phconcat _ _,
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let glue_adjoint := psusp_pelim (Y n) (Y (S n)) (glue Y n),
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exact pid X' ∧→ glue_adjoint,
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exact smash_functor_psquare (pvrefl f) (phrefl glue_adjoint),
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refine smash_functor_psquare (phrefl (pid X')) _,
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refine loop_to_susp_square _ _ _ _ _,
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exact smap.glue_square g n
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end
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/- smash of a spectrum and a type -/
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definition smash_spectrum (X : Type*) (Y : spectrum) : spectrum :=
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spectrify (smash_prespectrum X Y)
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definition smash_spectrum_fun {X X' : Type*} {Y Y' : spectrum} (f : X →* X') (g : Y →ₛ Y') : smash_spectrum X Y →ₛ smash_spectrum X' Y' :=
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spectrify_fun (smash_prespectrum_fun f g)
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end spectrum
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