Definition df-prstc 48864

Description: Definition of the function converting a preordered set to a category. Justified by prsthinc 48855.

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 48867, prstchom 48878, and prstcthin 48877. Other important properties include prstcbas 48868, prstcleval 48869, prstcle 48871, prstcocval 48872, prstcoc 48874, prstchom2 48879, and prstcprs 48876. Use those instead.

Note that the defining property prstchom 48878 is equivalent to prstchom2 48879 given prstcthin 48877. See thincn0eu 48832 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 48863	. 2 class ProsetToCat
2		vk	. . 3 setvar 𝑘
3		cproset 18350	. . 3 class Proset
4	2	cv 1536	. . . . 5 class 𝑘
5		cnx 17227	. . . . . . 7 class ndx
6		chom 17309	. . . . . . 7 class Hom
7	5, 6	cfv 6563	. . . . . 6 class (Hom ‘ndx)
8		cple 17305	. . . . . . . 8 class le
9	4, 8	cfv 6563	. . . . . . 7 class (le‘𝑘)
10		c1o 8498	. . . . . . . 8 class 1o
11	10	csn 4631	. . . . . . 7 class {1o}
12	9, 11	cxp 5687	. . . . . 6 class ((le‘𝑘) × {1o})
13	7, 12	cop 4637	. . . . 5 class ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩
14		csts 17197	. . . . 5 class sSet
15	4, 13, 14	co 7431	. . . 4 class (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩)
16		cco 17310	. . . . . 6 class comp
17	5, 16	cfv 6563	. . . . 5 class (comp‘ndx)
18		c0 4339	. . . . 5 class ∅
19	17, 18	cop 4637	. . . 4 class ⟨(comp‘ndx), ∅⟩
20	15, 19, 14	co 7431	. . 3 class ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩)
21	2, 3, 20	cmpt 5231	. 2 class (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
22	1, 21	wceq 1537	1 wff ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

This definition is referenced by:  prstcval  48865