Definition df-ran 50406

Description: Definition of the (local) right Kan extension. Given a functor 𝐹:𝐶⟶𝐷 and a functor 𝑋:𝐶⟶𝐸, the set (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) consists of right Kan extensions of 𝑋 along 𝐹, which are universal pairs from the pre-composition functor given by 𝐹 to 𝑋 (ranval2 50428). The definition in § 3 of Chapter X in p. 236 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf 50428 (retrieved 3 Nov 2025).

A right Kan extension is in the form of ⟨𝐿, 𝐴⟩ where the first component is a functor 𝐿:𝐷⟶𝐸 (ranrcl4 50437) and the second component is a natural transformation 𝐴:𝐿𝐹⟶𝑋 (ranrcl5 50438) where 𝐿𝐹 is the composed functor. Intuitively, the first component 𝐿 can be regarded as the result of an "inverse" of pre-composition; the source category of 𝑋:𝐶⟶𝐸 is "extended" along 𝐹:𝐶⟶𝐷.

The right Kan extension is a generalization of many categorical concepts such as limit. In § 7 of Chapter X of Categories for the Working Mathematician, it is concluded that "the notion of Kan extensions subsumes all the other fundamental concepts of category theory".

This definition was chosen over the other version in the commented out section due to its better reverse closure property.

See df-lan 50405 for the dual concept.

(Contributed by Zhi Wang, 4-Nov-2025.)

Assertion

Ref	Expression
df-ran	⊢ Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))

Distinct variable group:   𝑐,𝑑,𝑒,𝑓,𝑝,𝑥

Detailed syntax breakdown of Definition df-ran

Step	Hyp	Ref	Expression
1		cran 50404	. 2 class Ran
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3455	. . . 4 class V
5	4, 4	cxp 5659	. . 3 class (V × V)
6		vc	. . . 4 setvar 𝑐
7	2	cv 1569	. . . . 5 class 𝑝
8		c1st 7980	. . . . 5 class 1st
9	7, 8	cfv 6536	. . . 4 class (1st ‘𝑝)
10		vd	. . . . 5 setvar 𝑑
11		c2nd 7981	. . . . . 6 class 2nd
12	7, 11	cfv 6536	. . . . 5 class (2nd ‘𝑝)
13		vf	. . . . . 6 setvar 𝑓
14		vx	. . . . . 6 setvar 𝑥
15	6	cv 1569	. . . . . . 7 class 𝑐
16	10	cv 1569	. . . . . . 7 class 𝑑
17		cfunc 17906	. . . . . . 7 class Func
18	15, 16, 17	co 7410	. . . . . 6 class (𝑐 Func 𝑑)
19	3	cv 1569	. . . . . . 7 class 𝑒
20	15, 19, 17	co 7410	. . . . . 6 class (𝑐 Func 𝑒)
21	16, 19	cop 4595	. . . . . . . . 9 class ⟨𝑑, 𝑒⟩
22	13	cv 1569	. . . . . . . . 9 class 𝑓
23		cprcof 50171	. . . . . . . . 9 class −∘F
24	21, 22, 23	co 7410	. . . . . . . 8 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25		coppf 49920	. . . . . . . 8 class oppFunc
26	24, 25	cfv 6536	. . . . . . 7 class ( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))
27	14	cv 1569	. . . . . . 7 class 𝑥
28		cfuc 17997	. . . . . . . . . 10 class FuncCat
29	16, 19, 28	co 7410	. . . . . . . . 9 class (𝑑 FuncCat 𝑒)
30		coppc 17762	. . . . . . . . 9 class oppCat
31	29, 30	cfv 6536	. . . . . . . 8 class (oppCat‘(𝑑 FuncCat 𝑒))
32	15, 19, 28	co 7410	. . . . . . . . 9 class (𝑐 FuncCat 𝑒)
33	32, 30	cfv 6536	. . . . . . . 8 class (oppCat‘(𝑐 FuncCat 𝑒))
34		cup 49971	. . . . . . . 8 class UP
35	31, 33, 34	co 7410	. . . . . . 7 class ((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))
36	26, 27, 35	co 7410	. . . . . 6 class (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)
37	13, 14, 18, 20, 36	cmpo 7412	. . . . 5 class (𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥))
38	10, 12, 37	csb 3853	. . . 4 class ⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥))
39	6, 9, 38	csb 3853	. . 3 class ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥))
40	2, 3, 5, 4, 39	cmpo 7412	. 2 class (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))
41	1, 40	wceq 1570	1 wff Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))

This definition is referenced by:  ranfn  50408  reldmran  50410  ranfval  50412  relran  50422