Three years ago I thought it would be fun to write a blog post every day of November. I’m not sure why I didn’t do this in November 2010 or 2011 because I’m pretty sure I learned a lot from doing it in 2009, so I’d like to do it again. The posts will probably be shorter this time.

## Archive for October, 2012

## MaBloWriMo is upon us

Posted in remarks, tagged MaBloWriMo on October 28, 2012 | 3 Comments »

## Split epimorphisms and split monomorphisms

Posted in algebraic topology, category theory, group theory, topology, tagged abstract nonsense, universal properties on October 1, 2012 | 15 Comments »

There are various natural questions one can ask about monomorphisms and epimorphisms all of which lead to the same answer:

- What is the “easiest way” a morphism can be a monomorphism (resp. epimorphism)?
- What are the absolute monomorphisms (resp. epimorphisms) – that is, the ones which are preserved by every functor?
- A morphism which is both a monomorphism and an epimorphism is not necessarily an isomorphism. Can we replace either “monomorphism” or “epimorphism” by some other notion to repair this?
- If we wanted to generalize surjective functions, why didn’t we define an epimorphism to be a map which is surjective on generalized points?

The answer to all of these questions is the notion of a split monomorphism (resp. split epimorphism), which is the subject of today’s post.