Definition df-ran 49346

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⟩Ran𝐸)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 49366). 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 49366 (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 49374) and the second component is a natural transformation 𝐴:𝐿𝐹⟶𝑋 (ranrcl5 49375) where 𝐿𝐹 is the composed functor. Intuitively, the first component 𝐿 can be regarded as the result of a "inverse" of pre-composition; the source category is "extended" along 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 49345 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 49344	. 2 class Ran
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3457	. . . 4 class V
5	4, 4	cxp 5650	. . 3 class (V × V)
6		vc	. . . 4 setvar 𝑐
7	2	cv 1538	. . . . 5 class 𝑝
8		c1st 7981	. . . . 5 class 1st
9	7, 8	cfv 6528	. . . 4 class (1st ‘𝑝)
10		vd	. . . . 5 setvar 𝑑
11		c2nd 7982	. . . . . 6 class 2nd
12	7, 11	cfv 6528	. . . . 5 class (2nd ‘𝑝)
13		vf	. . . . . 6 setvar 𝑓
14		vx	. . . . . 6 setvar 𝑥
15	6	cv 1538	. . . . . . 7 class 𝑐
16	10	cv 1538	. . . . . . 7 class 𝑑
17		cfunc 17854	. . . . . . 7 class Func
18	15, 16, 17	co 7400	. . . . . 6 class (𝑐 Func 𝑑)
19	3	cv 1538	. . . . . . 7 class 𝑒
20	15, 19, 17	co 7400	. . . . . 6 class (𝑐 Func 𝑒)
21	16, 19	cop 4605	. . . . . . . . 9 class ⟨𝑑, 𝑒⟩
22	13	cv 1538	. . . . . . . . 9 class 𝑓
23		cprcof 49147	. . . . . . . . 9 class −∘F
24	21, 22, 23	co 7400	. . . . . . . 8 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25		coppf 48950	. . . . . . . 8 class oppFunc
26	24, 25	cfv 6528	. . . . . . 7 class (oppFunc‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))
27	14	cv 1538	. . . . . . 7 class 𝑥
28		cfuc 17945	. . . . . . . . . 10 class FuncCat
29	16, 19, 28	co 7400	. . . . . . . . 9 class (𝑑 FuncCat 𝑒)
30		coppc 17710	. . . . . . . . 9 class oppCat
31	29, 30	cfv 6528	. . . . . . . 8 class (oppCat‘(𝑑 FuncCat 𝑒))
32	15, 19, 28	co 7400	. . . . . . . . 9 class (𝑐 FuncCat 𝑒)
33	32, 30	cfv 6528	. . . . . . . 8 class (oppCat‘(𝑐 FuncCat 𝑒))
34		cup 48974	. . . . . . . 8 class UP
35	31, 33, 34	co 7400	. . . . . . 7 class ((oppCat‘(𝑑 FuncCat 𝑒))UP(oppCat‘(𝑐 FuncCat 𝑒)))
36	26, 27, 35	co 7400	. . . . . 6 class ((oppFunc‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒))UP(oppCat‘(𝑐 FuncCat 𝑒)))𝑥)
37	13, 14, 18, 20, 36	cmpo 7402	. . . . 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 3872	. . . 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 3872	. . 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 7402	. 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 1539	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  49348  reldmran  49350  ranfval  49352  relran  49360