[Alta-Logic] Talk by Richard Garner: 11:00am, 13th July 2011, ICT616

[Robin Cockett]

Who: Richard Garner
When: 11:00am Wed 13th July
Where: ICT616

Title: "On the axioms for adhesive and quasiadhesive categories"

(joint work with Steve Lack)

The notion of adhesive category was introduced by Lack and Sobocinski
as a formalisation of the basic structure on a category needed to
carry out graph rewriting in it. A category is adhesive if it has
finite limits and pushouts along monomorphisms, and these pushouts are
suitably well-behaved.

Unfortunately, in an adhesive category every monomorphism is
necessarily regular, and this means that many of the categories in
which graph-rewriters like to work  are not actually adhesive. On this
account Lack and Sobocinski also introduced the notion of
quasi-adhesive category; here one requires only well-behaved pushouts
along regular monomorphisms, and this allows an acceptably large class
of examples to be captured.

Any topos is adhesive and in fact (if we conveniently ignore size
issues) a category is adhesive if and only if it admits an embedding
into a topos which preserves the adhesive structure. Lack and
Sobocinski's intention was that quasi-adhesive categories should stand
in the same relationship to quasi-toposes, but unfortunately this is
not the case.

We shall discuss in this talk how the definition of quasi-adhesiveness
can be changed to patch this up. Along the way we will see that the
original definitions of adhesive and of quasi-adhesive
category---which are slightly fiddly---can be given in a much
simplified manner.

===============

Richard Garner obtained his PhD from the University of Cambridge in 
2006, he is currently a holder of an Australian Research Fellowship and 
works with the Sydney Category theory seminar at the University of 
Macquarie.  He has worked in higher dimensional category theory, 
homotopy theory, and type theory.