Definition df-prstc 47770

Description: Definition of the function converting a preordered set to a category. Justified by prsthinc 47761.

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 47773, prstchom 47784, and prstcthin 47783. Other important properties include prstcbas 47774, prstcleval 47775, prstcle 47777, prstcocval 47778, prstcoc 47780, prstchom2 47785, and prstcprs 47782. Use those instead.

Note that the defining property prstchom 47784 is equivalent to prstchom2 47785 given prstcthin 47783. See thincn0eu 47739 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 47769	. 2 class ProsetToCat
2		vk	. . 3 setvar 𝑘
3		cproset 18250	. . 3 class Proset
4	2	cv 1538	. . . . 5 class 𝑘
5		cnx 17130	. . . . . . 7 class ndx
6		chom 17212	. . . . . . 7 class Hom
7	5, 6	cfv 6542	. . . . . 6 class (Hom ‘ndx)
8		cple 17208	. . . . . . . 8 class le
9	4, 8	cfv 6542	. . . . . . 7 class (le‘𝑘)
10		c1o 8461	. . . . . . . 8 class 1o
11	10	csn 4627	. . . . . . 7 class {1o}
12	9, 11	cxp 5673	. . . . . 6 class ((le‘𝑘) × {1o})
13	7, 12	cop 4633	. . . . 5 class ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩
14		csts 17100	. . . . 5 class sSet
15	4, 13, 14	co 7411	. . . 4 class (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩)
16		cco 17213	. . . . . 6 class comp
17	5, 16	cfv 6542	. . . . 5 class (comp‘ndx)
18		c0 4321	. . . . 5 class ∅
19	17, 18	cop 4633	. . . 4 class ⟨(comp‘ndx), ∅⟩
20	15, 19, 14	co 7411	. . 3 class ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩)
21	2, 3, 20	cmpt 5230	. 2 class (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
22	1, 21	wceq 1539	1 wff ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

This definition is referenced by:  prstcval  47771