Can one define a "category of all mathematical objects"?

Can one define a "category of all mathematical objects"?

[Mattias Wikström]

Dear Categorists,

I would be interested in hearing what you think of an idea that seems
rather wild: Is it possible to define a category with nice properties
such that any locally small category becomes isomorphic to a
subcategory of that category?

In the absence of such a category, can a category such as the category
of all Grothendieck topoi and geometric morphisms between them serve
as a "category of all mathematical objects" for practical purposes?

Mattias Wikstrom

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Can one define a "category of all mathematical objects"?

[Dusko Pavlovic]

The Grothendieck Construction provides a somewhat trivial answer: a category fibered over Cat, where the fiber over each category is that category. But the object class of this fibered category does not live in the same universe as the object classes of the objects of Cat. Moreover, this construction is very redundant: eg each category A is disjoint from the category A+A, which consists of two copies of A. 

So the question is really: How can we glue together large families of categories in a less redundant way? For toposes, the answer seems ti be Gros Topos. Is this gluing business essentially topological, or are there other views?

just my 2c.

-- dusko


On Jul 23, 2010, at 12:49 AM, Mattias Wikström wrote:

> Dear Categorists,
> 
> I would be interested in hearing what you think of an idea that seems
> rather wild: Is it possible to define a category with nice properties
> such that any locally small category becomes isomorphic to a
> subcategory of that category?
> 
> In the absence of such a category, can a category such as the category
> of all Grothendieck topoi and geometric morphisms between them serve
> as a "category of all mathematical objects" for practical purposes?
> 
> Mattias Wikstrom
> 



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Can one define a "category of all mathematical objects"?

[Jiri Rosicky]

For concrete categories, this problem is addressed in the book
A. Pultr and V. Trnkova, Combinatorial, Algebraic and Topological
Representations of Groups, Semigroups and Categories, North-Holland 1980.
There is a locally small concrete category K containing all locally small
concrete categories as full subcategories (up to iso). Assuming
the non-existence of a proper class of measurable cardinals, we can
take the category Gra of graphs, or the category Smg of semigroups for K. 
Without this set-theoretic axiom, both Gra and Smg contain all accessible 
categories (i.e., all Grothendieck toposes) as full subcategories.
This can be found in my book (with J. Adamek) Locally Presentable 
and Accessible Categories, Cambridge Univ. Press 1994

----- Forwarded message from Mattias Wikström <mattias.wikstrom@gmail.com> -----

> Date: Fri, 23 Jul 2010 09:49:26 +0200
> From: Mattias Wikström <mattias.wikstrom@gmail.com>
> To: categories@mta.ca
> Subject: categories: Can one define a "category of all mathematical objects"?
> 
> Dear Categorists,
> 
> I would be interested in hearing what you think of an idea that seems
> rather wild: Is it possible to define a category with nice properties
> such that any locally small category becomes isomorphic to a
> subcategory of that category?
> 
> In the absence of such a category, can a category such as the category
> of all Grothendieck topoi and geometric morphisms between them serve
> as a "category of all mathematical objects" for practical purposes?
> 
> Mattias Wikstrom

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]