Subject: categories: Re: Categories ridiculously abstract

From: Michael MAKKAI <[email-masked]>
Date: Fri, 1 Dec 2000 17:19:59 -0500 (EST)

In "Towards a categorical foundation of mathematics" (Logic Colloquium
'95, ed's: J. A. Makowsky and E. V. Ravve, Springer Lecture Notes in Logic
no.11, 1998; pp.153-190) and in subsequent work, I am proposing an
approach to a foundation whose universe consists of the weak n-categories
and whatever things are needed to connect them. This is done on the basis
of a general point of view concerning the role of identity of mathematical
objects. Readers of said paper who have followed developments on weak
higher dimensional categories will note that much has been done since
towards fleshing out the program.

Michael Makkai

On Thu, 30 Nov 2000, Tom Leinster wrote:

> 
> Michael Barr wrote:
> > 
> > And here is a question: are categories more abstract or less abstract than
> > sets? 
> 
> A higher-dimensional category theorist's answer:
> "Neither - a set is merely a 0-category, and a category a 1-category."
> 
> There's a more serious thought behind this.  Sometimes I've wondered, in a
> vague way, whether the much-discussed hierarchy
> 
> 0-categories (sets) form a (1-)category, 
> (1-)categories form a 2-category, 
> ...
> 
> has a role to play in foundations.  After all, set-theorists seek to found
> mathematics on the theory of 0-categories; category-theorists sometimes talk
> about founding mathematics on the theory of 1-categories and providing a
> (Lawverian) axiomatization of the 1-category of 0-categories; you might ask
> "what next"?  Axiomatize the 2-category of (1-)categories?  Or the
> (n+1)-category of n-categories?  Could it even be, I ask with tongue in cheek
> and head in clouds, that general n-categories provide a more natural
> foundation than either 0-categories or 1-categories?
> 
> 
> Tom
>

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From: "Robert J. MacG. Dawson" <[email-masked]>
Date: Sat, 02 Dec 2000 09:34:22 -0400

Tom Leinster wrote:
> (n+1)-category of n-categories?  

	Surely we should start with the set of (-1)-categories? <gd&r>

	-Robert Dawson

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From: "DR Mawanda" <[email-masked]>
Date: Wed, 6 Dec 2000 21:18:24 +0200

My understanding of the relation between  category theory and  set  theory
is that category theory  is a formal theory built on abstract concepts
(objects and morphisms). The way of defining category theory need a
metalanguage which is closed to the logic of set theory language (a
particular case of what is called boolean logic). There is a sort of
dichothomy between logic behind the two theories. This dichothomy come from
our limitation of talking about category theory. We use  already two-valued
logic (true and false)  which we cannot avoid if we need to talk about
identity of objects and morphisms. Now a kind of Godel's arguments about
natural numbers (If N is consistent, then there is no proof of its
consistency by method formalizable within the theory ) is  what is going on.
This doesn't stop the category theory 'game'. When you give birth to a child
you will never know in advance if the child will be an honest person or a
criminal. Category theory have generated many structures which can help us
to understand why many mathematicians have work differently to describe a
same mathematical concept in different ways. As an example we know, from
category theory,  that Cauchy and Dedekind were defining real numbers from
rational numbers but the two definitions are not saying the same thing.