Definition df-mndtc 49567

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 49569), 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 49570, mndtchom 49573, mndtcco 49574. 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 49566	. 2 class MndToCat
2		vm	. . 3 setvar 𝑚
3		cmnd 18626	. . 3 class Mnd
4		cnx 17122	. . . . . 6 class ndx
5		cbs 17138	. . . . . 6 class Base
6	4, 5	cfv 6486	. . . . 5 class (Base‘ndx)
7	2	cv 1539	. . . . . 6 class 𝑚
8	7	csn 4579	. . . . 5 class {𝑚}
9	6, 8	cop 4585	. . . 4 class ⟨(Base‘ndx), {𝑚}⟩
10		chom 17190	. . . . . 6 class Hom
11	4, 10	cfv 6486	. . . . 5 class (Hom ‘ndx)
12	7, 5	cfv 6486	. . . . . . 7 class (Base‘𝑚)
13	7, 7, 12	cotp 4587	. . . . . 6 class ⟨𝑚, 𝑚, (Base‘𝑚)⟩
14	13	csn 4579	. . . . 5 class {⟨𝑚, 𝑚, (Base‘𝑚)⟩}
15	11, 14	cop 4585	. . . 4 class ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩
16		cco 17191	. . . . . 6 class comp
17	4, 16	cfv 6486	. . . . 5 class (comp‘ndx)
18	7, 7, 7	cotp 4587	. . . . . . 7 class ⟨𝑚, 𝑚, 𝑚⟩
19		cplusg 17179	. . . . . . . 8 class +g
20	7, 19	cfv 6486	. . . . . . 7 class (+g‘𝑚)
21	18, 20	cop 4585	. . . . . 6 class ⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩
22	21	csn 4579	. . . . 5 class {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}
23	17, 22	cop 4585	. . . 4 class ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩
24	9, 15, 23	ctp 4583	. . 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 1540	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  49568