34 lines
1.2 KiB
Text
34 lines
1.2 KiB
Text
import .direct_sum .quotient_group
|
||
|
||
open eq algebra is_trunc set_quotient relation sigma prod sum list trunc function equiv sigma.ops nat
|
||
|
||
namespace group
|
||
|
||
section
|
||
|
||
parameters (A : @trunctype.mk 0 ℕ _ → AddAbGroup) (f : Πi , A i → A (i + 1))
|
||
variables {A' : AddAbGroup}
|
||
|
||
definition seq_colim_carrier : AddAbGroup := dirsum A
|
||
inductive seq_colim_rel : seq_colim_carrier → Type :=
|
||
| rmk : Πi a, seq_colim_rel ((dirsum_incl A i a) - (dirsum_incl A (i + 1) (f i a)))
|
||
|
||
definition seq_colim : AddAbGroup := quotient_ab_group_gen seq_colim_carrier (λa, ∥seq_colim_rel a∥)
|
||
|
||
definition seq_colim_incl [constructor] (i : ℕ) : A i →a seq_colim :=
|
||
qg_map _ ∘g dirsum_incl A i
|
||
|
||
definition seq_colim_quotient (h : Πi, A i →a A') (k : Πi a, h i a = h (i + 1) (f i a))
|
||
(v : seq_colim_carrier) (r : ∥seq_colim_rel v∥) : dirsum_elim h v = 0 :=
|
||
begin
|
||
induction r with r, induction r,
|
||
end
|
||
|
||
definition seq_colim_elim [constructor] (h : Πi, A i →a A')
|
||
(k : Πi a, h i a = h (i + 1) (f i a)) : seq_colim →a A' :=
|
||
gqg_elim _ (dirsum_elim h) (seq_colim_quotient h k)
|
||
|
||
end
|
||
|
||
end group
|
||
|