On Adjointness of Comma Categories

Authors : M. IRFAN

Pages : 0-0

Doi:10.1501/Commua1_0000000104

View : 8 | Download : 8

Publication Date : 1981-01-01

Article Type : Research Paper

Abstract :In this paper we determine a SUMMARY pair of adjoint functors for the categories insert ignore into journalissuearticles values(T । A); and and insert ignore into journalissuearticles values(T` । A`); where T and T` are functors from to Q and A and A` are objects of A precise dcfinition of a Comma Category insert ignore into journalissuearticles values(T | A); where T is a functor from a category “g to another category Q and A is an obiect of g, in the sense of Maclane [2 ], can be stated as: The objects of insert ignore into journalissuearticles values(T | A); are pairs insert ignore into journalissuearticles values(B, b); with B h: insert ignore into journalissuearticles values(B,b); 1^1 andb:TB A and the morphisms insert ignore into journalissuearticles values(B`, b`); are those morphisms h: B -> B` in^ for ■which b`o Th = b. In the present note we determine a pair of adjoint functors for the categories insert ignore into journalissuearticles values(T | A); and insert ignore into journalissuearticles values(T` | A`); where, again. T` is a functor from “S to Q and A` is an object of g. Lemma 1: If a: T`-> T` is a natural transformation and u: A A` is a morphism, then for insert ignore into journalissuearticles values(B, b); in insert ignore into journalissuearticles values(T | A); the rule insert ignore into journalissuearticles values(B, b); defines a functor K: insert ignore into journalissuearticles values(T A); Proof: If we define R on insert ignore into journalissuearticles values(T` I A`); morphisms as: for any f: insert ignore into journalissuearticles values(B, b); insert ignore into journalissuearticles values(B, ub a B); insert ignore into journalissuearticles values(B`, b`); e R insert ignore into journalissuearticles values(f); = f such that ub`aB`oT` f = ub aB; that is, the following diagram commutes
Keywords : Adjointness, Comma Categories, Statistics