Definition df-retr 36104

Description: Define the set of retractions on two topological spaces. We say that 𝑅 is a retraction from 𝐽 to 𝐾. or 𝑅 ∈ (𝐽 Retr 𝐾) iff there is an 𝑆 such that 𝑅:𝐽⟶𝐾, 𝑆:𝐾⟶𝐽 are continuous functions called the retraction and section respectively, and their composite 𝑅 ∘ 𝑆 is homotopic to the identity map. If a retraction exists, we say 𝐽 is a retract of 𝐾. (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
df-retr	⊢ Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})

Distinct variable group:   𝑗,𝑘,𝑟,𝑠

Detailed syntax breakdown of Definition df-retr

Step	Hyp	Ref	Expression
1		cretr 36103	. 2 class Retr
2		vj	. . 3 setvar 𝑗
3		vk	. . 3 setvar 𝑘
4		ctop 23223	. . 3 class Top
5		vr	. . . . . . . . 9 setvar 𝑟
6	5	cv 1568	. . . . . . . 8 class 𝑟
7		vs	. . . . . . . . 9 setvar 𝑠
8	7	cv 1568	. . . . . . . 8 class 𝑠
9	6, 8	ccom 5658	. . . . . . 7 class (𝑟 ∘ 𝑠)
10		cid 5548	. . . . . . . 8 class I
11	2	cv 1568	. . . . . . . . 9 class 𝑗
12	11	cuni 4869	. . . . . . . 8 class ∪ 𝑗
13	10, 12	cres 5656	. . . . . . 7 class ( I ↾ ∪ 𝑗)
14		chtpy 25300	. . . . . . . 8 class Htpy
15	11, 11, 14	co 7427	. . . . . . 7 class (𝑗 Htpy 𝑗)
16	9, 13, 15	co 7427	. . . . . 6 class ((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗))
17		c0 4280	. . . . . 6 class ∅
18	16, 17	wne 2958	. . . . 5 wff ((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅
19	3	cv 1568	. . . . . 6 class 𝑘
20		ccn 23554	. . . . . 6 class Cn
21	19, 11, 20	co 7427	. . . . 5 class (𝑘 Cn 𝑗)
22	18, 7, 21	wrex 3089	. . . 4 wff ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅
23	11, 19, 20	co 7427	. . . 4 class (𝑗 Cn 𝑘)
24	22, 5, 23	crab 3415	. . 3 class {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅}
25	2, 3, 4, 4, 24	cmpo 7429	. 2 class (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})
26	1, 25	wceq 1569	1 wff Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})

This definition is used by: (None)