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 18661	. . 3 class Mnd
4		cnx 17163	. . . . . 6 class ndx
5		cbs 17179	. . . . . 6 class Base
6	4, 5	cfv 6511	. . . . 5 class (Base‘ndx)
7	2	cv 1539	. . . . . 6 class 𝑚
8	7	csn 4589	. . . . 5 class {𝑚}
9	6, 8	cop 4595	. . . 4 class ⟨(Base‘ndx), {𝑚}⟩
10		chom 17231	. . . . . 6 class Hom
11	4, 10	cfv 6511	. . . . 5 class (Hom ‘ndx)
12	7, 5	cfv 6511	. . . . . . 7 class (Base‘𝑚)
13	7, 7, 12	cotp 4597	. . . . . 6 class ⟨𝑚, 𝑚, (Base‘𝑚)⟩
14	13	csn 4589	. . . . 5 class {⟨𝑚, 𝑚, (Base‘𝑚)⟩}
15	11, 14	cop 4595	. . . 4 class ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩
16		cco 17232	. . . . . 6 class comp
17	4, 16	cfv 6511	. . . . 5 class (comp‘ndx)
18	7, 7, 7	cotp 4597	. . . . . . 7 class ⟨𝑚, 𝑚, 𝑚⟩
19		cplusg 17220	. . . . . . . 8 class +g
20	7, 19	cfv 6511	. . . . . . 7 class (+g‘𝑚)
21	18, 20	cop 4595	. . . . . 6 class ⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩
22	21	csn 4589	. . . . 5 class {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}
23	17, 22	cop 4595	. . . 4 class ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩
24	9, 15, 23	ctp 4593	. . 3 class {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩}
25	2, 3, 24	cmpt 5188	. 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