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

[Jiri Rosicky <rosicky@math.muni.cz>]

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

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