Factorization Structures and Natural Relations
by
Ryosuke Nakagawa

Let A be a complete category. For an A-object X, denote a product of X and X by S(X) and a diagonal morphism from X to S(X) by d(X). Then we have a functor S from A to A and a natural transformation d from the identity functor on A to the functor S. A natural relation R in A is a pair consisting of a functor R from A to A and a natural transformation j from R to S satisfying the following conditions.

(1) For each A-object X, j(X) is an extremal monomorphism. (2) There is a natural transformation i from the identity functor on A to R such that ji=d. For a natural relation R in A, an object X is said to be R-separated if i(X) is an isomorphism. It is known that if A is complete, cocomplete, well-powered and co-well-powered, the full subcategory of A consisting of all R-separated objects is an ExtrEpi-reflective subcategory of A. Let A be (E,M)-structured with classes E and M of A-morphisms. For an A-object X, let (R(X),i(X),j(X)) be an (E,M)-factorization of a diagonal morphism d(X). Then we have a natural relation R in A. In this talk we construct a factorization structure for morphisms in A from a natural relation R in A and discuss the correspondence between factorization structures and natural relations.

Date received: May 19, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-11.