Definition df-mndtc 49739

Description: Definition of the function converting a monoid to a category. Example 3.3(4.e) of [Adamek] p. 24.

The definition of the base set is arbitrary. The whole extensible structure becomes the object here (see mndtcbasval 49741), instead of just the base set, as is the case in Example 3.3(4.e) of [Adamek] p. 24.

The resulting category is defined entirely, up to isomorphism, by mndtcbas 49742, mndtchom 49745, mndtcco 49746. Use those instead.

See example 3.26(3) of [Adamek] p. 33 for more on isomorphism.

"MndToCat" was taken instead of "MndCat" because the latter might mean the category of monoids. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)

Assertion

Ref	Expression
df-mndtc	⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})

Detailed syntax breakdown of Definition df-mndtc

Step	Hyp	Ref	Expression
1		cmndtc 49738	. 2 class MndToCat
2		vm	. . 3 setvar 𝑚
3		cmnd 18650	. . 3 class Mnd
4		cnx 17111	. . . . . 6 class ndx
5		cbs 17127	. . . . . 6 class Base
6	4, 5	cfv 6489	. . . . 5 class (Base‘ndx)
7	2	cv 1540	. . . . . 6 class 𝑚
8	7	csn 4577	. . . . 5 class {𝑚}
9	6, 8	cop 4583	. . . 4 class ⟨(Base‘ndx), {𝑚}⟩
10		chom 17179	. . . . . 6 class Hom
11	4, 10	cfv 6489	. . . . 5 class (Hom ‘ndx)
12	7, 5	cfv 6489	. . . . . . 7 class (Base‘𝑚)
13	7, 7, 12	cotp 4585	. . . . . 6 class ⟨𝑚, 𝑚, (Base‘𝑚)⟩
14	13	csn 4577	. . . . 5 class {⟨𝑚, 𝑚, (Base‘𝑚)⟩}
15	11, 14	cop 4583	. . . 4 class ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩
16		cco 17180	. . . . . 6 class comp
17	4, 16	cfv 6489	. . . . 5 class (comp‘ndx)
18	7, 7, 7	cotp 4585	. . . . . . 7 class ⟨𝑚, 𝑚, 𝑚⟩
19		cplusg 17168	. . . . . . . 8 class +g
20	7, 19	cfv 6489	. . . . . . 7 class (+g‘𝑚)
21	18, 20	cop 4583	. . . . . 6 class ⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩
22	21	csn 4577	. . . . 5 class {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}
23	17, 22	cop 4583	. . . 4 class ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩
24	9, 15, 23	ctp 4581	. . 3 class {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩}
25	2, 3, 24	cmpt 5176	. 2 class (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
26	1, 25	wceq 1541	1 wff MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})

This definition is referenced by:  mndtcval  49740