2015-09-13 18:58:11 +00:00
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/-
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Copyright (c) 2015 Ulrik Buchholtz. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Ulrik Buchholtz
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-/
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2015-09-25 16:39:45 +00:00
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import types.trunc homotopy.sphere hit.pushout
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2015-09-13 18:58:11 +00:00
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open eq is_trunc is_equiv nat equiv trunc prod pushout sigma sphere_index unit
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-- where should this be?
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definition family : Type := ΣX, X → Type
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-- this should be in init!
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namespace nat
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definition cases {M : ℕ → Type} (mz : M zero) (ms : Πn, M (succ n)) : Πn, M n :=
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nat.rec mz (λn dummy, ms n)
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end nat
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namespace cellcomplex
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/-
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define by recursion on ℕ
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both the type of fdccs of dimension n
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and the realization map fdcc n → Type
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in other words, we define a function
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fdcc : ℕ → family
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an alternative to the approach here (perhaps necessary) is to
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define relative cell complexes relative to a type A, and then use
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spherical indexing, so a -1-dimensional relative cell complex is
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just star : unit with realization A
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-/
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definition fdcc_family [reducible] : ℕ → family :=
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nat.rec
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-- a zero-dimensional cell complex is just an hset
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-- with realization the identity map
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⟨hset , λA, trunctype.carrier A⟩
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(λn fdcc_family_n, -- sigma.rec (λ fdcc_n realize_n,
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/- a (succ n)-dimensional cell complex is a triple of
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an n-dimensional cell complex X, an hset of (succ n)-cells A,
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and an attaching map f : A × sphere n → |X| -/
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⟨Σ X : pr1 fdcc_family_n , Σ A : hset, A × sphere n → pr2 fdcc_family_n X ,
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/- the realization of such is the pushout of f with
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canonical map A × sphere n → unit -/
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sigma.rec (λX , sigma.rec (λA f, pushout (λx , star) f))
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⟩)
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definition fdcc (n : ℕ) : Type := pr1 (fdcc_family n)
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definition cell : Πn, fdcc n → hset :=
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nat.cases (λA, A) (λn T, pr1 (pr2 T))
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end cellcomplex
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