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Topic: Using category theory in my research
Replies: 2   Last Post: Oct 30, 2013 9:41 AM

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@less@ndro

Posts: 215
Registered: 12/13/04
Re: Using category theory in my research
Posted: Oct 30, 2013 9:41 AM
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Victor Porton <porton@narod.ru> wrote:
> I have constructed products, coproducts, equalizers and coequalizers of some
> categories which appeared in my research:
>
> http://www.mathematics21.org/algebraic-general-topology.html
>
> It follows that my categories are complete and cocomplete.
>
> What I as a category theory novice should do next?
>
> Should I provide explicit formulas for pullbacks and pushouts?
>
> I have also tried to prove that my categories is are cartesian closed, but
> without success as for now.
>
> As my categories are concrete, I will attempt to prove that the forgetful
> functors have both left and right adjoints.
>
> What else should I do, based on category theoretic ideology?


I have not yet found time to take a closer look at your manuscript,
so I may miss a lot of points completely. But as you claim to provide
a new and better framework for studying topology and uniformity, it
might be interesting, how the usual categories of topological or
uniform spaces (also e.g. locales) relate to your categories.

Picture yourself in the situation of a topologist who might be interested
in your approach but does not want to throw all his results away. Perhaps
there are some nice functors between the old categories and your new ones,
perhaps equivalences between various subcategories, perhaps some "copy"
of some nice spaces within your new categories etc.

As I said above, since I have not read your manuscript, I am not at all
specific . But, in order not to end up in isolation, it would certainly
help if you could relate your new stuff to those things that are already
there. If you do so, you may find that category theory is not an ideology
but a very useful conceptual tool.

P.S.: for a mansucript of this length an index would be very useful.

--
Marc




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