Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 208:a1e5d2a3d3bd
fix
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 02 Sep 2013 17:13:14 +0900 |
parents | 22811f7a04e1 |
children | 4e138cc953f3 |
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205 | 1 --- |
2 -- | |
3 -- Equalizer | |
4 -- | |
208 | 5 -- e f |
205 | 6 -- c --------> a ----------> b |
208 | 7 -- ^ . ----------> |
205 | 8 -- | . g |
208 | 9 -- |k . |
10 -- | . h | |
205 | 11 -- d |
12 -- | |
13 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
14 ---- | |
15 | |
16 open import Category -- https://github.com/konn/category-agda | |
17 open import Level | |
18 open import Category.Sets | |
19 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where | |
20 | |
21 open import HomReasoning | |
22 open import cat-utility | |
23 | |
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24 record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
205 | 25 field |
208 | 26 equalizer : {c d : Obj A} (e : Hom A c a) (h : Hom A d a) → Hom A d c |
27 equalize : {c d : Obj A} (e : Hom A c a) (h : Hom A d a) → | |
28 A [ A [ A [ f o e ] o equalizer e h ] ≈ A [ g o h ] ] | |
29 uniqueness : {c d : Obj A} (e : Hom A c a) (h : Hom A d a) ( k : Hom A d c ) → | |
30 A [ A [ A [ f o e ] o k ] ≈ A [ g o h ] ] → A [ equalizer e h ≈ k ] | |
206 | 31 |
207
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32 record EqEqualizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
206 | 33 field |
207
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34 α : {e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → Hom A e a |
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35 γ : {c d e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A c e |
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36 δ : {e a b : Obj A} → (f : Hom A a b) → Hom A a e |
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37 b1 : {e : Obj A} → A [ A [ f o α {e} f g ] ≈ A [ g o α {e} f g ] ] |
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38 b2 : {c d : Obj A } → {h : Hom A d a } → A [ A [ α {c} f g o γ {c} f g h ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) ] ] |
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39 b3 : {e : Obj A} → A [ A [ α {e} f f o δ {e} f ] ≈ id1 A a ] |
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40 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
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41 b4 : {c d : Obj A } {k : Hom A c a} → A [ A [ γ f g ( A [ α f g o k ] ) o δ {c} (A [ f o A [ α f g o k ] ] ) ] ≈ k ] |
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42 -- A [ α f g o β f g h ] ≈ h |
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43 β : { d e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d e |
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44 β {d} f g h = A [ γ f g h o δ {d} (A [ f o h ]) ] |
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45 |
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46 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} (f g : Hom A a b) → Equalizer A f g → EqEqualizer A f g |
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47 lemma-equ1 A {a} {b} f g eqa = record { |
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48 α = {!!} ; |
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49 γ = {!!} ; |
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50 δ = {!!} ; |
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51 b1 = {!!} ; |
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52 b2 = {!!} ; |
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53 b3 = {!!} ; |
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54 b4 = {!!} |
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55 } |