Definition df-ran

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 ). 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 (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 ) and the second component is a natural transformation A : L F --> X ( ranrcl5 ) where L F 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 ofCategories 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 for the dual concept.

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

	Ref	Expression
Assertion	df-ran	Could not format assertion : No typesetting found for |- Ran = ( p e. ( _V X. _V ) , e e. _V |-> [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) ) with typecode |-

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cran	Could not format Ran : No typesetting found for class Ran with typecode class
1		vp	$$setvarp$$
2		cvv	$$class\mathrm{V}$$
3	2 2	cxp	$$class\left(\mathrm{V}\times \mathrm{V}\right)$$
4		ve	$$setvare$$
5		c1st	$$class{1}^{st}$$
6	1	cv	$$setvarp$$
7	6 5	cfv	$$class{1}^{st}\left(p\right)$$
8		vc	$$setvarc$$
9		c2nd	$$class{2}^{nd}$$
10	6 9	cfv	$$class{2}^{nd}\left(p\right)$$
11		vd	$$setvard$$
12		vf	$$setvarf$$
13	8	cv	$$setvarc$$
14		cfunc	$$class\mathrm{Func}$$
15	11	cv	$$setvard$$
16	13 15 14	co	$$class\left(c\mathrm{Func}d\right)$$
17		vx	$$setvarx$$
18	4	cv	$$setvare$$
19	13 18 14	co	$$class\left(c\mathrm{Func}e\right)$$
20		coppf	Could not format oppFunc : No typesetting found for class oppFunc with typecode class
21	15 18	cop	$$class⟨d,e⟩$$
22		cprcof	Could not format -o.F : No typesetting found for class -o.F with typecode class
23	12	cv	$$setvarf$$
24	21 23 22	co	Could not format ( <. d , e >. -o.F f ) : No typesetting found for class ( <. d , e >. -o.F f ) with typecode class
25	24 20	cfv	Could not format ( oppFunc ` ( <. d , e >. -o.F f ) ) : No typesetting found for class ( oppFunc ` ( <. d , e >. -o.F f ) ) with typecode class
26		coppc	$$class\mathrm{oppCat}$$
27		cfuc	$$class\mathrm{FuncCat}$$
28	15 18 27	co	$$class\left(d\mathrm{FuncCat}e\right)$$
29	28 26	cfv	$$class\mathrm{oppCat}\left(d\mathrm{FuncCat}e\right)$$
30		cup	Could not format UP : No typesetting found for class UP with typecode class
31	13 18 27	co	$$class\left(c\mathrm{FuncCat}e\right)$$
32	31 26	cfv	$$class\mathrm{oppCat}\left(c\mathrm{FuncCat}e\right)$$
33	29 32 30	co	Could not format ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) : No typesetting found for class ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) with typecode class
34	17	cv	$$setvarx$$
35	25 34 33	co	Could not format ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) : No typesetting found for class ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) with typecode class
36	12 17 16 19 35	cmpo	Could not format ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) : No typesetting found for class ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) with typecode class
37	11 10 36	csb	Could not format [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) : No typesetting found for class [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) with typecode class
38	8 7 37	csb	Could not format [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) : No typesetting found for class [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) with typecode class
39	1 4 3 2 38	cmpo	Could not format ( p e. ( _V X. _V ) , e e. _V |-> [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) ) : No typesetting found for class ( p e. ( _V X. _V ) , e e. _V |-> [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) ) with typecode class
40	0 39	wceq	Could not format Ran = ( p e. ( _V X. _V ) , e e. _V |-> [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) ) : No typesetting found for wff Ran = ( p e. ( _V X. _V ) , e e. _V |-> [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) ) with typecode wff