Frederik Hanghøj Iversen
7aec22b30a
Expose both monad formulations qualified from Cat.Category.Monad
2018-03-14 11:00:52 +01:00
Frederik Hanghøj Iversen
6229decfb2
Merge branch 'master' into dev
2018-03-14 10:50:57 +01:00
Frederik Hanghøj Iversen
41e2d02c8d
[WIP] Prove voe §2.3
...
By Andrea
The reason you cannot use cong in [1] is that §2-fromMonad result type
depends on the input, you need a dependent version of cong:
cong-d : ∀ {ℓ} {A : Set ℓ} {ℓ'} {B : A → Set ℓ'} {x y : A}
→ (f : (x : A) → B x)
→ (eq : x ≡ y)
→ PathP (\ i → B (eq i)) (f x) (f y)
cong-d f p = λ i → f (p i)
I attach a modified Voevodsky.agda.
Notice that the definition of "t" is still highlighted in yellow,
that's because it being a homogeneous path depends on the exact
definition of lem, see the comment with the two definitional equality
constraints.
2018-03-14 10:30:42 +01:00
Frederik Hanghøj Iversen
091e77b583
Rename IsProduct.isProduct to IsProduct.ump
...
[WIP]: Also some stuff about propositionality for products.
2018-03-14 10:23:23 +01:00
Frederik Hanghøj Iversen
7065455712
More readable goal for voevodsky's construction
2018-03-13 11:29:13 +01:00
Frederik Hanghøj Iversen
896e0d3d37
Stuff about univalence for the category of functors
2018-03-13 10:24:50 +01:00
Frederik Hanghøj Iversen
fe453a6d3a
Trying to prove cummulativity of homotopy levels
2018-03-12 16:00:27 +01:00
Frederik Hanghøj Iversen
c52384b012
Change name of fromMonad
2018-03-12 14:43:43 +01:00
Frederik Hanghøj Iversen
5e092964c8
Change naming and fuse some modules
2018-03-12 14:38:52 +01:00
Frederik Hanghøj Iversen
ccf753d438
Move monoidal and kleisli representation to own modules
2018-03-12 14:23:23 +01:00
Frederik Hanghøj Iversen
8dadfa22a0
Add documentation header to monad module
2018-03-12 14:11:31 +01:00
Frederik Hanghøj Iversen
aa645fb11e
Move voevodsky's construction to own module
2018-03-12 14:04:10 +01:00
Frederik Hanghøj Iversen
c0cf6789cd
Use propositions straight from the horses mouth
2018-03-12 13:56:49 +01:00
Frederik Hanghøj Iversen
a7214fcc66
Finish equality principle for categories
2018-03-12 13:51:29 +01:00
Frederik Hanghøj Iversen
35390c02d3
Stuff about univalence in the category of sets
2018-03-12 13:38:48 +01:00
Frederik Hanghøj Iversen
acb5ff4f2b
Closer to showing univalence for the category of sets
2018-03-08 14:44:23 +01:00
Frederik Hanghøj Iversen
52297d9073
Clean-up in the category of categories
2018-03-08 11:54:13 +01:00
Frederik Hanghøj Iversen
d01514cbdb
Do not use ugly ':'-syntax to disambiguate fields
2018-03-08 11:29:16 +01:00
Frederik Hanghøj Iversen
48672b01bd
Use dotted expression in Cat
2018-03-08 11:20:51 +01:00
Frederik Hanghøj Iversen
5ad506a09f
Rename func* and func-> to omap and fmap respectively
2018-03-08 11:03:56 +01:00
Frederik Hanghøj Iversen
2fcc583646
Add note
2018-03-08 10:50:18 +01:00
Frederik Hanghøj Iversen
63b5f5c68d
Use long name for product object
2018-03-08 10:46:28 +01:00
Frederik Hanghøj Iversen
486238e114
Add goals for propositionality of products
2018-03-08 10:38:46 +01:00
Frederik Hanghøj Iversen
1ef57a19f4
Cosmetics
2018-03-08 10:30:35 +01:00
Frederik Hanghøj Iversen
4e7b350188
Factor out objects
2018-03-08 10:28:05 +01:00
Frederik Hanghøj Iversen
181bd1af53
Factor out category
2018-03-08 10:24:17 +01:00
Frederik Hanghøj Iversen
faf4c54188
Make parameters explicit
2018-03-08 10:22:21 +01:00
Frederik Hanghøj Iversen
fae492a1e3
Restructure products
2018-03-08 10:20:29 +01:00
Frederik Hanghøj Iversen
b61749bb91
Fixup some todo-notes
2018-03-08 01:10:52 +01:00
Frederik Hanghøj Iversen
fa9a470875
Update backlog
2018-03-08 00:54:42 +01:00
Frederik Hanghøj Iversen
e43bee6d9f
Feels really close
2018-03-08 00:36:38 +01:00
Frederik Hanghøj Iversen
c8fef1d2b5
Use different name for function composition
2018-03-08 00:22:55 +01:00
Frederik Hanghøj Iversen
36cbe711fb
Sort of half of the proof of an inverse
2018-03-08 00:09:49 +01:00
Frederik Hanghøj Iversen
459718da23
Finish proof of equivalence of klesili/monoidal categories!!
2018-03-07 17:30:09 +01:00
Frederik Hanghøj Iversen
19103e1678
Update cubical
2018-03-07 16:24:43 +01:00
Frederik Hanghøj Iversen
93d075a6d3
Attempt at proving pureNTEq
2018-03-07 15:23:07 +01:00
Frederik Hanghøj Iversen
00e6e1aa66
State problem with approach
2018-03-07 11:45:11 +01:00
Frederik Hanghøj Iversen
aa64e01084
Remove some cruft
2018-03-07 11:33:08 +01:00
Frederik Hanghøj Iversen
125123846e
Lay out a strategy for showing the equivalence
2018-03-07 11:29:58 +01:00
Frederik Hanghøj Iversen
085e6eb3d7
Stuff about voe-2-3
2018-03-06 23:18:33 +01:00
Frederik Hanghøj Iversen
110e3510c5
Use postulates
2018-03-06 15:55:03 +01:00
Frederik Hanghøj Iversen
5ae68df582
Prove that fmap is mapped correctly
2018-03-06 15:53:11 +01:00
Frederik Hanghøj Iversen
4d528a7077
Clean-up
2018-03-06 11:25:29 +01:00
Frederik Hanghøj Iversen
485703c85e
Tidy up
2018-03-06 10:16:42 +01:00
Frederik Hanghøj Iversen
0cebe1e866
Make private
2018-03-06 10:06:45 +01:00
Frederik Hanghøj Iversen
4de27aa06c
Naming
2018-03-06 10:05:35 +01:00
Frederik Hanghøj Iversen
9173468b03
Use omap/fmap
2018-03-06 09:56:44 +01:00
Frederik Hanghøj Iversen
bdd67aee53
Rename RR to Romap
2018-03-06 09:55:18 +01:00
Frederik Hanghøj Iversen
c57cd5c991
Define stuff in monoidal record
2018-03-06 09:52:37 +01:00
Frederik Hanghøj Iversen
cfb7925cb5
Renaming
2018-03-06 09:45:04 +01:00
Frederik Hanghøj Iversen
b6457a0b14
Add comment
2018-03-06 09:41:29 +01:00
Frederik Hanghøj Iversen
7647a452cd
Tidy up proof a bit
2018-03-06 09:39:48 +01:00
Frederik Hanghøj Iversen
35419ad86e
Rename eta and mu
2018-03-06 09:35:50 +01:00
Frederik Hanghøj Iversen
f8e08288a0
Cosmetics
2018-03-05 17:31:13 +01:00
Frederik Hanghøj Iversen
9ec6ce9eba
Use other equality principle
2018-03-05 17:10:41 +01:00
Frederik Hanghøj Iversen
3151fb3e46
Prove propositionality for naturality
2018-03-05 16:35:47 +01:00
Frederik Hanghøj Iversen
7f4a8a65b8
More stuff about opposite being an involution
2018-03-05 16:10:27 +01:00
Frederik Hanghøj Iversen
b26ea18257
Cleanup in nattrans
2018-03-05 15:04:16 +01:00
Frederik Hanghøj Iversen
ddd5f17c05
Move propositionality stuff about natural transformations to that module
2018-03-05 15:02:36 +01:00
Frederik Hanghøj Iversen
2b92cee254
Prettier names in Fun
2018-03-05 14:55:45 +01:00
Frederik Hanghøj Iversen
bb379fa196
Implement category of presheaves
2018-03-05 14:50:53 +01:00
Frederik Hanghøj Iversen
ce4dd83969
Prove that the yoneda embedding is distributive
2018-03-05 14:42:12 +01:00
Frederik Hanghøj Iversen
7fbca1aeeb
Clean-up yoneda embedding
2018-03-05 14:04:04 +01:00
Frederik Hanghøj Iversen
1bf565b87a
Have yoneda without having a category of categories
...
I did break some things in Cat.Categories.Cat but since this is
unprovable anyways it's not that big a deal.
2018-03-05 13:52:59 +01:00
Frederik Hanghøj Iversen
5c3616bca5
Make argument to presheaf explicit
2018-03-05 11:17:31 +01:00
Frederik Hanghøj Iversen
059c74b687
Use already defined category
2018-03-05 11:15:45 +01:00
Frederik Hanghøj Iversen
a4890a42cf
Define Monoidal categories without depending on category of categories
2018-03-05 11:13:58 +01:00
Frederik Hanghøj Iversen
5902c6121b
Further reduce dependency on impossible facts.
...
Provide the data for the product in the category of categories without
requiring such a category to actually exist
2018-03-05 11:07:42 +01:00
Frederik Hanghøj Iversen
77006011d3
Minimize dependency on category of categories
2018-03-05 10:35:33 +01:00
Frederik Hanghøj Iversen
8f8800cb67
More stuff about kleisli \equiv monoidal
2018-03-05 10:28:16 +01:00
Frederik Hanghøj Iversen
b079f5e426
Prove propositionality for IsMonad
2018-03-02 13:31:46 +01:00
Frederik Hanghøj Iversen
c4e3625746
Finish proof of distributivity
2018-03-01 20:47:36 +01:00
Frederik Hanghøj Iversen
2ceb027f7a
Prove monad-equality principle for kleisly monads
2018-03-01 20:23:34 +01:00
Frederik Hanghøj Iversen
f2164a6717
Prove equality principle for monads
2018-03-01 20:12:49 +01:00
Frederik Hanghøj Iversen
a7f31bb3e2
Prove "foreign naturality condition"
2018-03-01 18:00:51 +01:00
Frederik Hanghøj Iversen
f526fd6010
Prove inverse law
2018-03-01 17:50:06 +01:00
Frederik Hanghøj Iversen
ff2952e9ad
Make postulate
2018-03-01 14:59:19 +01:00
Frederik Hanghøj Iversen
ae46a48861
Define goals in Kleisli
2018-03-01 14:58:01 +01:00
Frederik Hanghøj Iversen
64a0292755
Cosmetics
2018-03-01 14:19:46 +01:00
Frederik Hanghøj Iversen
e8b29e1f7f
\mu is join and it's a natural transformation!
2018-02-28 23:41:59 +01:00
Frederik Hanghøj Iversen
9d3b17245f
Provide \zeta
2018-02-28 19:32:07 +01:00
Frederik Hanghøj Iversen
f2b1a36a75
Define and use Endofunctor
2018-02-28 19:03:11 +01:00
Frederik Hanghøj Iversen
3c77c69cf6
Move functor definition to Kleisli.Monad
2018-02-28 19:00:21 +01:00
Frederik Hanghøj Iversen
70221377d3
Move proof of equivalence to IsMonad making them lemmas
2018-02-28 18:55:32 +01:00
Frederik Hanghøj Iversen
1aaf81552c
Move another proof to category definition
2018-02-26 20:42:00 +01:00
Frederik Hanghøj Iversen
101b2639e1
Move proof to category definition
2018-02-26 20:31:47 +01:00
Frederik Hanghøj Iversen
5b5d21f777
Formatting
2018-02-26 20:23:31 +01:00
Frederik Hanghøj Iversen
a0944d69b1
Documentation in Monad
2018-02-26 20:08:48 +01:00
Frederik Hanghøj Iversen
67993be27b
Add reverse function composition to category
2018-02-26 20:00:24 +01:00
Frederik Hanghøj Iversen
47882b1110
Rename zeta to pure
2018-02-26 19:58:27 +01:00
Frederik Hanghøj Iversen
043641462d
Prove distributive law for monads!
2018-02-26 19:57:05 +01:00
Frederik Hanghøj Iversen
7cddba97a8
Shorten definition
2018-02-25 19:03:48 +01:00
Frederik Hanghøj Iversen
2c6132768e
Remove Pathy and Bij
2018-02-25 15:29:52 +01:00
Frederik Hanghøj Iversen
5caecf9796
Rename properties to yoneda
2018-02-25 15:28:42 +01:00
Frederik Hanghøj Iversen
44526b85eb
Move CwF
2018-02-25 15:24:44 +01:00
Frederik Hanghøj Iversen
f0beec1530
Rename Opposite to opposite
2018-02-25 15:23:33 +01:00
Frederik Hanghøj Iversen
cd98736d02
Add documentation in Category-module
2018-02-25 15:21:38 +01:00
Frederik Hanghøj Iversen
2e7220567a
Move lemma into IsCategory
2018-02-25 14:44:03 +01:00
Frederik Hanghøj Iversen
d63ecc3a65
Use abbreviation
2018-02-25 14:39:11 +01:00
Frederik Hanghøj Iversen
caddf83a09
Let IsCategory reexport RawCategory
2018-02-25 14:37:28 +01:00
Frederik Hanghøj Iversen
5deabb7546
Forgot to add monoid-module
2018-02-25 14:28:01 +01:00
Frederik Hanghøj Iversen
ce46e0ae7a
Module-ify
2018-02-25 14:27:37 +01:00
Frederik Hanghøj Iversen
12dddc2067
Use a module
2018-02-25 03:12:51 +01:00
Frederik Hanghøj Iversen
4c298855e0
[WIP] Proving other fusion law
...
Also set up framework for equality principle for monads
2018-02-25 03:09:25 +01:00
Frederik Hanghøj Iversen
a6b01929f0
Prove distributive law
2018-02-25 01:27:20 +01:00
Frederik Hanghøj Iversen
a447cd9c7c
Syntax
2018-02-24 20:41:47 +01:00
Frederik Hanghøj Iversen
9d09363f78
Expand definition of isDistributive somewhat
...
Also contains some side-tracks
2018-02-24 20:37:21 +01:00
Frederik Hanghøj Iversen
e7abab0e4c
Add pure and >=> to kleisli category
2018-02-24 19:08:20 +01:00
Frederik Hanghøj Iversen
be505cdfbe
Prove IsAssociative
2018-02-24 19:07:58 +01:00
Frederik Hanghøj Iversen
5d9c820fa2
Add note about haskell
2018-02-24 15:25:07 +01:00
Frederik Hanghøj Iversen
e4e327d1d2
[WIP] equivalence of kleisli- resp. monoidal- representation of monad
2018-02-24 15:13:25 +01:00
Frederik Hanghøj Iversen
3e12331294
Monoidal monads addendum
2018-02-24 14:01:57 +01:00
Frederik Hanghøj Iversen
4ec13fe509
Implement monads in the kleisli form
2018-02-24 14:00:52 +01:00
Frederik Hanghøj Iversen
0ca11874bc
Remove old name for functor composition
2018-02-24 12:55:08 +01:00
Frederik Hanghøj Iversen
8527fe0df4
Rename functor composition - implement monads...
...
In their monoidal form.
2018-02-24 12:52:16 +01:00
Frederik Hanghøj Iversen
cb8533b84a
Rename natural transformation composition
2018-02-23 17:43:38 +01:00
Frederik Hanghøj Iversen
dd11b69c71
Documentation for natural transformations
2018-02-23 17:37:27 +01:00
Frederik Hanghøj Iversen
689a6467c6
Move stuff about natural transformations to own module
2018-02-23 17:33:09 +01:00
Frederik Hanghøj Iversen
f5dded9561
Do not use IsCategory directly
2018-02-23 16:41:17 +01:00
Frederik Hanghøj Iversen
39284b8d99
Changes in CwF
2018-02-23 14:13:55 +01:00
Frederik Hanghøj Iversen
5796b791b8
Almost prove that arrows are sets in the cateogry of families
2018-02-23 13:59:35 +01:00
Frederik Hanghøj Iversen
a321a9c8b2
Use hLevels in Fam
2018-02-23 13:39:59 +01:00
Frederik Hanghøj Iversen
3d0916f448
Use correct name for hSets
2018-02-23 13:20:30 +01:00
Frederik Hanghøj Iversen
4874ed0795
Rename distrib to isDistributive
2018-02-23 12:53:35 +01:00
Frederik Hanghøj Iversen
7787a8f0be
Indentation
2018-02-23 12:52:14 +01:00
Frederik Hanghøj Iversen
48423cc816
Rename arrowIsSet to arrowsAreSets
2018-02-23 12:51:44 +01:00
Frederik Hanghøj Iversen
6446435a49
Rename ident to isIdentity
2018-02-23 12:49:41 +01:00
Frederik Hanghøj Iversen
5cbc409770
Rename assoc to isAssociative
2018-02-23 12:43:49 +01:00
Frederik Hanghøj Iversen
852056cc44
Add type-synonyms in functor
2018-02-23 12:41:15 +01:00
Frederik Hanghøj Iversen
a57f45d93f
Remove yet another postulate
2018-02-23 12:33:20 +01:00
Frederik Hanghøj Iversen
34dec9406d
Do not mention IsFunctor outside the module that defines it
2018-02-23 12:29:10 +01:00
Frederik Hanghøj Iversen
e46edf1f68
Chain reexport things in Functor
2018-02-23 12:21:16 +01:00
Frederik Hanghøj Iversen
885fd8fa69
Drastically simplify proofs
2018-02-23 12:15:39 +01:00
Frederik Hanghøj Iversen
3f3247c870
Remove commented code
2018-02-23 12:05:38 +01:00
Frederik Hanghøj Iversen
de1d19c442
Readd stuff about the yoneda embedding
2018-02-23 11:24:22 +01:00
Frederik Hanghøj Iversen
954a89f8d1
Expose naturalIsProp
2018-02-23 11:12:27 +01:00
Frederik Hanghøj Iversen
bc2129b8fc
Readd yoneda embedding
2018-02-23 10:55:43 +01:00
Frederik Hanghøj Iversen
9a4d79fa4e
Readd commented code
2018-02-23 10:44:23 +01:00
Frederik Hanghøj Iversen
3032dc6130
Make explicit argument
2018-02-23 10:36:59 +01:00
Frederik Hanghøj Iversen
cc1ddaac9f
Add new type-synonym
2018-02-23 10:35:42 +01:00
Frederik Hanghøj Iversen
a87d404aad
Refactor category of categories
...
No longer actually define the category. Just define the raw category and
a few results about it.
2018-02-23 10:34:37 +01:00
Frederik Hanghøj Iversen
32b9ce2ea8
Use new syntax in cat
2018-02-22 15:31:54 +01:00
Frederik Hanghøj Iversen
a82095604d
Remove unused function
2018-02-21 14:05:10 +01:00
Frederik Hanghøj Iversen
9e96e704e8
Update Fun according to new naming policy
2018-02-21 13:40:24 +01:00
Frederik Hanghøj Iversen
57d7eab4cb
Make sets a category according to HoTT
2018-02-21 13:37:07 +01:00
Frederik Hanghøj Iversen
ed40824edc
Cosmetics
2018-02-21 12:59:31 +01:00
Frederik Hanghøj Iversen
edf552cb86
Do not define synonym for contractible
2018-02-20 18:15:30 +01:00
Frederik Hanghøj Iversen
d2da84269f
Move some more things into RawCategory
2018-02-20 18:14:42 +01:00
Frederik Hanghøj Iversen
0c861c4bde
Factor univalence out to a seperate module
2018-02-20 18:13:06 +01:00
Frederik Hanghøj Iversen
a4f8a37e36
Proove that IsCategory is a mere proposition!
2018-02-20 18:01:26 +01:00
