On Adjointness of Comma Categories

Year 1981, Volume: 30 , 0 - 0, 01.01.1981

M. Irfan

https://doi.org/10.1501/Commua1_0000000104

Abstract

In this paper we determine a
SUMMARY
pair of adjoint functors for the categories (T । A) and and (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 (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 (T | A) are pairs (B, b) with B h: (B,b)
1^1 andb:TB A and the morphisms (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 (T | A) and (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 (B, b) in (T | A) the rule (B, b) defines a functor K: (T A) Proof: If we define R on
(T' I A')
morphisms as: for any f: (B, b)
(B, ub a B)
(B', b')
e
R (f) = f such that ub'aB'oT' f = ub aB; that is, the following diagram commutes

Keywords

Adjointness, Comma Categories, Statistics

References

• Ankara Üniversitesi – Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Dergisi