Definition df-ran 49590

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 49612). 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 49612 (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 49621) and the second component is a natural transformation 𝐴:𝐿𝐹⟶𝑋 (ranrcl5 49622) 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 49589 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 49588	. 2 class Ran
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3444	. . . 4 class V
5	4, 4	cxp 5629	. . 3 class (V × V)
6		vc	. . . 4 setvar 𝑐
7	2	cv 1539	. . . . 5 class 𝑝
8		c1st 7945	. . . . 5 class 1st
9	7, 8	cfv 6499	. . . 4 class (1st ‘𝑝)
10		vd	. . . . 5 setvar 𝑑
11		c2nd 7946	. . . . . 6 class 2nd
12	7, 11	cfv 6499	. . . . 5 class (2nd ‘𝑝)
13		vf	. . . . . 6 setvar 𝑓
14		vx	. . . . . 6 setvar 𝑥
15	6	cv 1539	. . . . . . 7 class 𝑐
16	10	cv 1539	. . . . . . 7 class 𝑑
17		cfunc 17796	. . . . . . 7 class Func
18	15, 16, 17	co 7369	. . . . . 6 class (𝑐 Func 𝑑)
19	3	cv 1539	. . . . . . 7 class 𝑒
20	15, 19, 17	co 7369	. . . . . 6 class (𝑐 Func 𝑒)
21	16, 19	cop 4591	. . . . . . . . 9 class ⟨𝑑, 𝑒⟩
22	13	cv 1539	. . . . . . . . 9 class 𝑓
23		cprcof 49355	. . . . . . . . 9 class −∘F
24	21, 22, 23	co 7369	. . . . . . . 8 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25		coppf 49104	. . . . . . . 8 class oppFunc
26	24, 25	cfv 6499	. . . . . . 7 class ( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))
27	14	cv 1539	. . . . . . 7 class 𝑥
28		cfuc 17887	. . . . . . . . . 10 class FuncCat
29	16, 19, 28	co 7369	. . . . . . . . 9 class (𝑑 FuncCat 𝑒)
30		coppc 17652	. . . . . . . . 9 class oppCat
31	29, 30	cfv 6499	. . . . . . . 8 class (oppCat‘(𝑑 FuncCat 𝑒))
32	15, 19, 28	co 7369	. . . . . . . . 9 class (𝑐 FuncCat 𝑒)
33	32, 30	cfv 6499	. . . . . . . 8 class (oppCat‘(𝑐 FuncCat 𝑒))
34		cup 49155	. . . . . . . 8 class UP
35	31, 33, 34	co 7369	. . . . . . 7 class ((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))
36	26, 27, 35	co 7369	. . . . . 6 class (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)
37	13, 14, 18, 20, 36	cmpo 7371	. . . . 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 3859	. . . 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 3859	. . 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 7371	. 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 1540	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  49592  reldmran  49594  ranfval  49596  relran  49606