Definition df-prstc 47173

Description: Definition of the function converting a preordered set to a category. Justified by prsthinc 47164.

This definition is somewhat arbitrary. Example 3.3(4.d) of [Adamek] p. 24 demonstrates an alternate definition with pairwise disjoint hom-sets. The behavior of the function is defined entirely, up to isomorphism, by prstcnid 47176, prstchom 47187, and prstcthin 47186. Other important properties include prstcbas 47177, prstcleval 47178, prstcle 47180, prstcocval 47181, prstcoc 47183, prstchom2 47188, and prstcprs 47185. Use those instead.

Note that the defining property prstchom 47187 is equivalent to prstchom2 47188 given prstcthin 47186. See thincn0eu 47142 for justification.

"ProsetToCat" was taken instead of "ProsetCat" because the latter might mean the category of preordered sets (classes). However, "ProsetToCat" seems too long. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)

Assertion

Ref	Expression
df-prstc	⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

Detailed syntax breakdown of Definition df-prstc

Step	Hyp	Ref	Expression
1		cprstc 47172	. 2 class ProsetToCat
2		vk	. . 3 setvar 𝑘
3		cproset 18190	. . 3 class Proset
4	2	cv 1541	. . . . 5 class 𝑘
5		cnx 17073	. . . . . . 7 class ndx
6		chom 17152	. . . . . . 7 class Hom
7	5, 6	cfv 6500	. . . . . 6 class (Hom ‘ndx)
8		cple 17148	. . . . . . . 8 class le
9	4, 8	cfv 6500	. . . . . . 7 class (le‘𝑘)
10		c1o 8409	. . . . . . . 8 class 1o
11	10	csn 4590	. . . . . . 7 class {1o}
12	9, 11	cxp 5635	. . . . . 6 class ((le‘𝑘) × {1o})
13	7, 12	cop 4596	. . . . 5 class ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩
14		csts 17043	. . . . 5 class sSet
15	4, 13, 14	co 7361	. . . 4 class (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩)
16		cco 17153	. . . . . 6 class comp
17	5, 16	cfv 6500	. . . . 5 class (comp‘ndx)
18		c0 4286	. . . . 5 class ∅
19	17, 18	cop 4596	. . . 4 class ⟨(comp‘ndx), ∅⟩
20	15, 19, 14	co 7361	. . 3 class ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩)
21	2, 3, 20	cmpt 5192	. 2 class (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
22	1, 21	wceq 1542	1 wff ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

This definition is referenced by:  prstcval  47174