Definition df-ran 49967

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 49989). 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 49989 (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 49998) and the second component is a natural transformation 𝐴:𝐿𝐹⟶𝑋 (ranrcl5 49999) 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 49966 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 49965	. 2 class Ran
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3442	. . . 4 class V
5	4, 4	cxp 5630	. . 3 class (V × V)
6		vc	. . . 4 setvar 𝑐
7	2	cv 1541	. . . . 5 class 𝑝
8		c1st 7941	. . . . 5 class 1st
9	7, 8	cfv 6500	. . . 4 class (1st ‘𝑝)
10		vd	. . . . 5 setvar 𝑑
11		c2nd 7942	. . . . . 6 class 2nd
12	7, 11	cfv 6500	. . . . 5 class (2nd ‘𝑝)
13		vf	. . . . . 6 setvar 𝑓
14		vx	. . . . . 6 setvar 𝑥
15	6	cv 1541	. . . . . . 7 class 𝑐
16	10	cv 1541	. . . . . . 7 class 𝑑
17		cfunc 17790	. . . . . . 7 class Func
18	15, 16, 17	co 7368	. . . . . 6 class (𝑐 Func 𝑑)
19	3	cv 1541	. . . . . . 7 class 𝑒
20	15, 19, 17	co 7368	. . . . . 6 class (𝑐 Func 𝑒)
21	16, 19	cop 4588	. . . . . . . . 9 class ⟨𝑑, 𝑒⟩
22	13	cv 1541	. . . . . . . . 9 class 𝑓
23		cprcof 49732	. . . . . . . . 9 class −∘F
24	21, 22, 23	co 7368	. . . . . . . 8 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25		coppf 49481	. . . . . . . 8 class oppFunc
26	24, 25	cfv 6500	. . . . . . 7 class ( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))
27	14	cv 1541	. . . . . . 7 class 𝑥
28		cfuc 17881	. . . . . . . . . 10 class FuncCat
29	16, 19, 28	co 7368	. . . . . . . . 9 class (𝑑 FuncCat 𝑒)
30		coppc 17646	. . . . . . . . 9 class oppCat
31	29, 30	cfv 6500	. . . . . . . 8 class (oppCat‘(𝑑 FuncCat 𝑒))
32	15, 19, 28	co 7368	. . . . . . . . 9 class (𝑐 FuncCat 𝑒)
33	32, 30	cfv 6500	. . . . . . . 8 class (oppCat‘(𝑐 FuncCat 𝑒))
34		cup 49532	. . . . . . . 8 class UP
35	31, 33, 34	co 7368	. . . . . . 7 class ((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))
36	26, 27, 35	co 7368	. . . . . 6 class (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)
37	13, 14, 18, 20, 36	cmpo 7370	. . . . 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 3851	. . . 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 3851	. . 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 7370	. 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 1542	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  49969  reldmran  49971  ranfval  49973  relran  49983