# Cochains

Cochains are diagrams of the form X₀ ← X₁ ← ⋯.
Author: Langston Barrett (@siddharthist), Febuary 2018
Define the cochain:
```     0 <-- 1 <-- 2 <-- 3 <-- ...
```
with exactly one arrow from S n to n.

Definition conat_graph : graph :=
make_graph nat (λ m n, S n = m).

Notation "'cochain'" := (diagram conat_graph).

A diagram for a cochain is what it should be, a collection of objects and arrows arranged so: X₀ ⟵ X₁ ⟵ ⋯. This can be used to easily construct cochains, see e.g. termCochain.
Definition cochain_weq {C : precategory} :
( (obs : n : nat, ob C), ( n : nat, obs (S n) --> obs n)) cochain C.
Proof.
use weqfibtototal; intro obs; cbn.
use weq_iso.
- intros ars a b aeqSb.
refine (_ · ars b).
exact (transportf (λ o, C obs o, obs (S b) ) aeqSb (identity _)).
- exact (λ ars n, ars (S n) n (idpath _)).
- intros ars; cbn; unfold idfun.
apply funextsec; intro n.
apply id_left.
- intros ars.
cbn.
apply funextsec; intro a.
apply funextsec; intro b.
apply funextsec; intro p.
induction p.
cbn; unfold idfun.
apply id_left.
Defined.

Definition mapcochain {C D : precategory} (F : functor C D)
(c : cochain C) : cochain D := mapdiagram F c.

Any j > i gives a morphism in the cochain via composition
Definition cochain_mor {C : precategory} (c : cochain C) {i j} :
i < j Cdob c j, dob c i.
Proof.
induction j as [|j IHj].
- intros Hi0; destruct (negnatlthn0 0 Hi0).
- intros Hij.
destruct (natlehchoice4 _ _ Hij) as [|H].
+ refine (_ · IHj h).
apply (dmor c), (idpath _).
+ apply (dmor c), (maponpaths S H).
Defined.

Construct the cochain:
```         !          F!            F^2 !
1 <----- F 1 <------ F^2 1 <-------- F^3 1 <--- ...
```
Definition termCochain {C : precategory} (TermC : Terminal C) (F : functor C C) :
cochain C.
Proof.
use cochain_weq; use tpair.
- exact (λ m, iter_functor F m TermC).
- intros n; induction n as [|n IHn].
× exact (TerminalArrow TermC _).
× exact (# F IHn).
Defined.