Definition df-mndtc 50234

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 50236), 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 50237, mndtchom 50240, mndtcco 50241. 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 50233	. 2 class MndToCat
2		vm	. . 3 setvar 𝑚
3		cmnd 18788	. . 3 class Mnd
4		cnx 17249	. . . . . 6 class ndx
5		cbs 17265	. . . . . 6 class Base
6	4, 5	cfv 6533	. . . . 5 class (Base‘ndx)
7	2	cv 1566	. . . . . 6 class 𝑚
8	7	csn 4591	. . . . 5 class {𝑚}
9	6, 8	cop 4597	. . . 4 class ⟨(Base‘ndx), {𝑚}⟩
10		chom 17317	. . . . . 6 class Hom
11	4, 10	cfv 6533	. . . . 5 class (Hom ‘ndx)
12	7, 5	cfv 6533	. . . . . . 7 class (Base‘𝑚)
13	7, 7, 12	cotp 4599	. . . . . 6 class ⟨𝑚, 𝑚, (Base‘𝑚)⟩
14	13	csn 4591	. . . . 5 class {⟨𝑚, 𝑚, (Base‘𝑚)⟩}
15	11, 14	cop 4597	. . . 4 class ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩
16		cco 17318	. . . . . 6 class comp
17	4, 16	cfv 6533	. . . . 5 class (comp‘ndx)
18	7, 7, 7	cotp 4599	. . . . . . 7 class ⟨𝑚, 𝑚, 𝑚⟩
19		cplusg 17306	. . . . . . . 8 class +g
20	7, 19	cfv 6533	. . . . . . 7 class (+g‘𝑚)
21	18, 20	cop 4597	. . . . . 6 class ⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩
22	21	csn 4591	. . . . 5 class {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}
23	17, 22	cop 4597	. . . 4 class ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩
24	9, 15, 23	ctp 4595	. . 3 class {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩}
25	2, 3, 24	cmpt 5193	. 2 class (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
26	1, 25	wceq 1567	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  50235