Definition df-ran 50930

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 50952). 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 50952 (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 50961) and the second component is a natural transformation 𝐴:𝐿𝐹⟶𝑋 (ranrcl5 50962) 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 50929 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 50928	. 2 class Ran
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3453	. . . 4 class V
5	4, 4	cxp 5652	. . 3 class (V × V)
6		vc	. . . 4 setvar 𝑐
7	2	cv 1568	. . . . 5 class 𝑝
8		c1st 8006	. . . . 5 class 1st
9	7, 8	cfv 6546	. . . 4 class (1st ‘𝑝)
10		vd	. . . . 5 setvar 𝑑
11		c2nd 8007	. . . . . 6 class 2nd
12	7, 11	cfv 6546	. . . . 5 class (2nd ‘𝑝)
13		vf	. . . . . 6 setvar 𝑓
14		vx	. . . . . 6 setvar 𝑥
15	6	cv 1568	. . . . . . 7 class 𝑐
16	10	cv 1568	. . . . . . 7 class 𝑑
17		cfunc 18041	. . . . . . 7 class Func
18	15, 16, 17	co 7427	. . . . . 6 class (𝑐 Func 𝑑)
19	3	cv 1568	. . . . . . 7 class 𝑒
20	15, 19, 17	co 7427	. . . . . 6 class (𝑐 Func 𝑒)
21	16, 19	cop 4837	. . . . . . . . 9 class ⟨𝑑, 𝑒⟩
22	13	cv 1568	. . . . . . . . 9 class 𝑓
23		cprcof 50647	. . . . . . . . 9 class −∘F
24	21, 22, 23	co 7427	. . . . . . . 8 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25		coppf 50377	. . . . . . . 8 class oppFunc
26	24, 25	cfv 6546	. . . . . . 7 class ( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))
27	14	cv 1568	. . . . . . 7 class 𝑥
28		cfuc 18132	. . . . . . . . . 10 class FuncCat
29	16, 19, 28	co 7427	. . . . . . . . 9 class (𝑑 FuncCat 𝑒)
30		coppc 17897	. . . . . . . . 9 class oppCat
31	29, 30	cfv 6546	. . . . . . . 8 class (oppCat‘(𝑑 FuncCat 𝑒))
32	15, 19, 28	co 7427	. . . . . . . . 9 class (𝑐 FuncCat 𝑒)
33	32, 30	cfv 6546	. . . . . . . 8 class (oppCat‘(𝑐 FuncCat 𝑒))
34		cup 50428	. . . . . . . 8 class UP
35	31, 33, 34	co 7427	. . . . . . 7 class ((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))
36	26, 27, 35	co 7427	. . . . . 6 class (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)
37	13, 14, 18, 20, 36	cmpo 7429	. . . . 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 3850	. . . 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 3850	. . 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 7429	. 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 1569	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 used by:  ranfn  50932  reldmran  50934  ranfval  50936  relran  50946