Maps of prespectra
Content created by Fredrik Bakke and Egbert Rijke.
Created on 2023-10-12.
Last modified on 2024-04-25.
module synthetic-homotopy-theory.maps-of-prespectra where
Imports
open import elementary-number-theory.natural-numbers open import foundation.dependent-pair-types open import foundation.universe-levels open import structured-types.commuting-squares-of-pointed-maps open import structured-types.pointed-maps open import synthetic-homotopy-theory.functoriality-loop-spaces open import synthetic-homotopy-theory.prespectra
Idea
A map of prespectra f : A → B
is a
sequence of
pointed maps
fₙ : Aₙ →∗ Bₙ
such that the squares
fₙ
Aₙ --------> Bₙ
| |
| |
| |
∨ ∨
ΩAₙ₊₁ -----> ΩBₙ₊₁
Ωfₙ₊₁
commute in the category of pointed types.
Definition
coherence-map-Prespectrum : {l1 l2 : Level} (n : ℕ) (A : Prespectrum l1) (B : Prespectrum l2) → ( (n : ℕ) → pointed-type-Prespectrum A n →∗ pointed-type-Prespectrum B n) → UU (l1 ⊔ l2) coherence-map-Prespectrum n A B f = coherence-square-pointed-maps ( f n) ( pointed-adjoint-structure-map-Prespectrum A n) ( pointed-adjoint-structure-map-Prespectrum B n) ( pointed-map-Ω (f (succ-ℕ n))) map-Prespectrum : {l1 l2 : Level} (A : Prespectrum l1) (B : Prespectrum l2) → UU (l1 ⊔ l2) map-Prespectrum A B = Σ ( (n : ℕ) → pointed-type-Prespectrum A n →∗ pointed-type-Prespectrum B n) ( λ f → (n : ℕ) → coherence-map-Prespectrum n A B f)
References
- [May99]
- J. P. May. A Concise Course in Algebraic Topology. University of Chicago Press, 09 1999. ISBN 978-0-226-51183-2. URL: https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf.
Recent changes
- 2024-04-25. Fredrik Bakke. chore: Fix arrowheads in character diagrams (#1124).
- 2024-03-12. Fredrik Bakke. Bibliographies (#1058).
- 2023-10-16. Fredrik Bakke and Egbert Rijke. Sequential limits (#839).
- 2023-10-12. Fredrik Bakke. Define suspension prespectra and maps of prespectra (#833).