Definition df-prstc 48069

Description: Definition of the function converting a preordered set to a category. Justified by prsthinc 48060.

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 48072, prstchom 48083, and prstcthin 48082. Other important properties include prstcbas 48073, prstcleval 48074, prstcle 48076, prstcocval 48077, prstcoc 48079, prstchom2 48084, and prstcprs 48081. Use those instead.

Note that the defining property prstchom 48083 is equivalent to prstchom2 48084 given prstcthin 48082. See thincn0eu 48038 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 48068	. 2 class ProsetToCat
2		vk	. . 3 setvar 𝑘
3		cproset 18285	. . 3 class Proset
4	2	cv 1533	. . . . 5 class 𝑘
5		cnx 17162	. . . . . . 7 class ndx
6		chom 17244	. . . . . . 7 class Hom
7	5, 6	cfv 6548	. . . . . 6 class (Hom ‘ndx)
8		cple 17240	. . . . . . . 8 class le
9	4, 8	cfv 6548	. . . . . . 7 class (le‘𝑘)
10		c1o 8480	. . . . . . . 8 class 1o
11	10	csn 4629	. . . . . . 7 class {1o}
12	9, 11	cxp 5676	. . . . . 6 class ((le‘𝑘) × {1o})
13	7, 12	cop 4635	. . . . 5 class ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩
14		csts 17132	. . . . 5 class sSet
15	4, 13, 14	co 7420	. . . 4 class (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩)
16		cco 17245	. . . . . 6 class comp
17	5, 16	cfv 6548	. . . . 5 class (comp‘ndx)
18		c0 4323	. . . . 5 class ∅
19	17, 18	cop 4635	. . . 4 class ⟨(comp‘ndx), ∅⟩
20	15, 19, 14	co 7420	. . 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 1534	1 wff ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

This definition is referenced by:  prstcval  48070