Re: Explanations

["Mattias Wikström" <mattias.wikstrom@gmail.com>]

> On 26/04/2011 06:55, Timothy Porter wrote:
> In category theory, many proofs are transparent and of the form: what do
> we know about the situation, just one fact, so we have to use that....
> it works. (I am thinking of classical Yoneda lemma type situations,
> since the only elements in hom-sets that we can be sure exist are the
> identities.) [...]

The question is: "What facts and what objects do we have?", "What is
given?". In general we need to work with what is directly given in
order to arrive at some indirectly given thing that we are seeking
(what we are seeking has to be given in some sense, or else the
problem cannot be solved).

In category-theoretic terms we can think of what is given as a
subobject A of some larger object B in an allegory (where B represents
things that "exist" but which we may or may not be able to refer to).
A subobject of B is then given just in case it factors through A. (For
this to work well the allegory we are working with should contain lots
of different objects so that "subobject of X" and "part of X" become
practically synonymous.)

We may also view things in terms of symmetry and invariants. We are
directly given certain subobjects A1, A2, ..., An of some object B in
an allegory and we are indirectly given any subobject of B which stays
invariant as we apply isomorphisms to B that fix A1, A2, ..., An. (The
earlier object A would be the smallest subobject of B containing A1,
A2, ..., An as parts.)

Finally, we can view what we are given as a theory (axiom system), and
the question is what can be defined/specified/referred to in that
theory. Of course, mathematics inevitably involves axiom systems, but
any theorem which starts "for all ..." can be thought of as involving
an axiom system of its own.

Mattias

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