Require Import Subcategories_ch7_1.
Require Import NaturalTransformations_ch7_4.
Require Import NaturalIsomorphisms_ch7_5.
+Require Import PreMonoidalCategories.
Require Import MonoidalCategories_ch7_8.
Require Import Coherence_ch7_8.
Require Import Enrichment_ch2_8.
+Require Import Enrichments.
Require Import RepresentableStructure_ch7_2.
Require Import Reification.
Require Import WeakFunctorCategory.
-Require Import SmallSMMEs.
-(* Technically reifications form merely a *semicategory* (no identity
+(*
+ * Technically reifications form merely a *semicategory* (no identity
* maps), but one can always freely adjoin identity maps (and nothing
* else) to a semicategory to get a category whose non-identity-map
- * portion is identical to the original semicategory
+ * portion is identical to the original semicategory (closing under
+ * composition after putting in the identity maps never produces any
+ * additional maps)
*
* Also, technically this category has ALL enrichments (not just the
* surjective monic monoidal ones), though there maps OUT OF only the
* use it as a host language. But that's for the next paper...
*)
Inductive ReificationOrIdentity : SMMEs -> SMMEs -> Type :=
- | roi_id : forall smme:SMMEs, ReificationOrIdentity smme smme
- | roi_reif : forall s1 s2:SMMEs, Reification s1 s2 (mon_i s2) -> ReificationOrIdentity s1 s2.
+ | roi_id : forall smme:SMMEs, ReificationOrIdentity smme smme
+ | roi_reif : forall s1 s2:SMMEs, Reification s1 s2 (enr_c_i s2) -> ReificationOrIdentity s1 s2.
Definition reificationOrIdentityFobj s1 s2 (f:ReificationOrIdentity s1 s2) : s1 -> s2 :=
match f with
Defined.
Definition compose_reifications (s0 s1 s2:SMMEs) :
- Reification s0 s1 (mon_i s1) -> Reification s1 s2 (mon_i s2) -> Reification s0 s2 (mon_i s2).
+ Reification s0 s1 (enr_c_i s1) -> Reification s1 s2 (enr_c_i s2) -> Reification s0 s2 (enr_c_i s2).
intros.
refine
{| reification_rstar_f := reification_rstar_f X >>>> reification_rstar_f X0
- ; reification_rstar := MonoidalFunctorsCompose _ _ _ _ _ (reification_rstar X) (reification_rstar X0)
- ; reification_r := fun K => (reification_r X K) >>>> (reification_r X0 (mon_i s1))
+ ; reification_rstar := PreMonoidalFunctorsCompose (reification_rstar X) (reification_rstar X0)
+ ; reification_r := fun K => (reification_r X K) >>>> (reification_r X0 (enr_c_i s1))
|}.
intro K.
- set (ni_associativity (reification_r X K) (reification_r X0 (mon_i s1)) (RepresentableFunctor s2 (mon_i s2))) as q.
+ set (ni_associativity (reification_r X K) (reification_r X0 (enr_c_i s1)) (HomFunctor s2 (enr_c_i s2))) as q.
eapply ni_comp.
apply q.
clear q.
set (reification_commutes X K) as comm1.
- set (reification_commutes X0 (mon_i s1)) as comm2.
- set (RepresentableFunctor s0 K) as a in *.
+ set (reification_commutes X0 (enr_c_i s1)) as comm2.
+ set (HomFunctor s0 K) as a in *.
set (reification_rstar_f X) as a' in *.
set (reification_rstar_f X0) as x in *.
set (reification_r X K) as b in *.
- set (reification_r X0 (mon_i s1)) as c in *.
- set (RepresentableFunctor s2 (mon_i s2)) as c' in *.
- set (RepresentableFunctor s1 (mon_i s1)) as b' in *.
+ set (reification_r X0 (enr_c_i s1)) as c in *.
+ set (HomFunctor s2 (enr_c_i s2)) as c' in *.
+ set (HomFunctor s1 (enr_c_i s1)) as b' in *.
apply (ni_comp(F2:=b >>>> (b' >>>> x))).
- apply (@ni_respects _ _ _ _ _ _ _ _ _ _ b _ b _ (c >>>> c') _ (b' >>>> x)).
- apply ni_id.
+ apply (ni_respects1 b (c >>>> c') (b' >>>> x)).
apply comm2.
eapply ni_comp.
eapply ni_inv.
eapply ni_comp.
eapply ni_inv.
apply (ni_associativity a a' x).
- apply (@ni_respects _ _ _ _ _ _ _ _ _ _ (a >>>> a') _ (b >>>> b') _ x _ x).
+ apply (ni_respects2 (a >>>> a') (b >>>> b') x).
apply ni_inv.
apply comm1.
- apply ni_id.
+ apply (reification_host_lang_pure _ _ _ X0).
Defined.
Definition reificationOrIdentityComp