Maths - Category Theory - Relation

So far on these pages we have looked at forms of equality between categories, here we look at how 'equalities' within objects might be represented.

Epimorphism (Surjective) Functions

In category theory we treat objects and functions as black boxes, don't look inside them, but here we will look inside just to get some intuition.

In set theory surjective mappings allow multiple elements in the domain to map to a single element in the codomain.

One possible way to interpret this is that all the elements that map to the same element are somehow related - equal in some way or have common properties.

Reversing the function gives a fibre bundle.

reverse function

More Information

Pullback

The above idea of surjective mappings identifying elements that are somehow equal can be further extended using a pullback.

P is the pullback of f along g or g along f (A,B,C fixed) .

Square must commute (in best possible way) so any other square, also containing (A,B,C) must uniquely map to it.

pullback

'P' will contain pairs of elements which are somehow 'equal'. That is all possible equations.

More Information

Topoi

More information on this page.


metadata block
see also:
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 

Terminology and Notation

Specific to this page here:

 

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2021 Martin John Baker - All rights reserved - privacy policy.