Definition df-prstc 45850

Description: Definition of the function converting a preordered set to a category. Justified by prsthinc 45846.

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 45853, prstchom 45862, and prstcthin 45861. Other important properties include prstcbas 45854, prstcleval 45855, prstcle 45856, prstcocval 45857, prstcoc 45858, prstchom2 45863, and prstcprs 45860. Use those instead.

Note that the defining property prstchom 45862 is equivalent to prstchom2 45863 given prstcthin 45861. See thincn0eu 45836 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 45849	. 2 class ProsetToCat
2		vk	. . 3 setvar 𝑘
3		cproset 17664	. . 3 class Proset
4	2	cv 1541	. . . . 5 class 𝑘
5		cnx 16595	. . . . . . 7 class ndx
6		chom 16691	. . . . . . 7 class Hom
7	5, 6	cfv 6349	. . . . . 6 class (Hom ‘ndx)
8		cple 16687	. . . . . . . 8 class le
9	4, 8	cfv 6349	. . . . . . 7 class (le‘𝑘)
10		c1o 8136	. . . . . . . 8 class 1o
11	10	csn 4526	. . . . . . 7 class {1o}
12	9, 11	cxp 5533	. . . . . 6 class ((le‘𝑘) × {1o})
13	7, 12	cop 4532	. . . . 5 class ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩
14		csts 16596	. . . . 5 class sSet
15	4, 13, 14	co 7182	. . . 4 class (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩)
16		cco 16692	. . . . . 6 class comp
17	5, 16	cfv 6349	. . . . 5 class (comp‘ndx)
18		c0 4221	. . . . 5 class ∅
19	17, 18	cop 4532	. . . 4 class ⟨(comp‘ndx), ∅⟩
20	15, 19, 14	co 7182	. . 3 class ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩)
21	2, 3, 20	cmpt 5120	. 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  45851