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/ ]
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/ ]
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/ ]
participants (3)
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Dusko Pavlovic -
Jiri Rosicky -
Mattias Wikström