Definition df-mndtc 45979

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 45981) , 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 45982, mndtchom 45985, mndtcco 45986. 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 45978	. 2 class MndToCat
2		vm	. . 3 setvar 𝑚
3		cmnd 18127	. . 3 class Mnd
4		cnx 16663	. . . . . 6 class ndx
5		cbs 16666	. . . . . 6 class Base
6	4, 5	cfv 6358	. . . . 5 class (Base‘ndx)
7	2	cv 1542	. . . . . 6 class 𝑚
8	7	csn 4527	. . . . 5 class {𝑚}
9	6, 8	cop 4533	. . . 4 class ⟨(Base‘ndx), {𝑚}⟩
10		chom 16760	. . . . . 6 class Hom
11	4, 10	cfv 6358	. . . . 5 class (Hom ‘ndx)
12	7, 5	cfv 6358	. . . . . . 7 class (Base‘𝑚)
13	7, 7, 12	cotp 4535	. . . . . 6 class ⟨𝑚, 𝑚, (Base‘𝑚)⟩
14	13	csn 4527	. . . . 5 class {⟨𝑚, 𝑚, (Base‘𝑚)⟩}
15	11, 14	cop 4533	. . . 4 class ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩
16		cco 16761	. . . . . 6 class comp
17	4, 16	cfv 6358	. . . . 5 class (comp‘ndx)
18	7, 7, 7	cotp 4535	. . . . . . 7 class ⟨𝑚, 𝑚, 𝑚⟩
19		cplusg 16749	. . . . . . . 8 class +g
20	7, 19	cfv 6358	. . . . . . 7 class (+g‘𝑚)
21	18, 20	cop 4533	. . . . . 6 class ⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩
22	21	csn 4527	. . . . 5 class {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}
23	17, 22	cop 4533	. . . 4 class ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩
24	9, 15, 23	ctp 4531	. . 3 class {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩}
25	2, 3, 24	cmpt 5120	. 2 class (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
26	1, 25	wceq 1543	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  45980