What else do simplicial sets classify?

What else do simplicial sets classify?

[Andrej Bauer]

The presheaf category of simplicial sets is the classifying topos for
the theory L of a bounded linear order.

In general, there could be other theories which are "Morita
equivalent" to L in the sense that their classifying toposes are
equivalent to simplicial sets. Are any such known, preferably
occurring in nature?

With kind regards,

Andrej


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Re: What else do simplicial sets classify?

[Prof. Peter Johnstone]

On Sat, 31 Jul 2010, Andrej Bauer wrote:

> The presheaf category of simplicial sets is the classifying topos for
> the theory L of a bounded linear order.
>
> In general, there could be other theories which are "Morita
> equivalent" to L in the sense that their classifying toposes are
> equivalent to simplicial sets. Are any such known, preferably
> occurring in nature?
>
> With kind regards,
>
> Andrej
>
Of course you can write down different presentations for the theory
classified by simplicial sets; any set of generators for the topos
will give you one. But you're unlikely to find anything familiar: if
there were a geometric theory "occurring in nature" whose category of
models (in any topos) was equivalent to the category of bounded linear
orders, that equivalence would almost certainly be well known.

Peter Johnstone



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What else do simplicial sets classify?

[Joyal, André]

Dear Andrej,

I do not know what you mean by a bounded linear order.

But I know that the topos of simplicial sets
is classifying strict intervals.

(an interval [a,b] is strict if its endpoint a and b are different)

  bounded linear orders = strict intervals?

Best, 
André

-------- Message d'origine--------
De: Andrej Bauer [mailto:andrej.bauer@andrej.com]
Date: sam. 31/07/2010 03:55
À: categories list
Objet : categories: What else do simplicial sets classify?
 
The presheaf category of simplicial sets is the classifying topos for
the theory L of a bounded linear order.

In general, there could be other theories which are "Morita
equivalent" to L in the sense that their classifying toposes are
equivalent to simplicial sets. Are any such known, preferably
occurring in nature?

With kind regards,

Andrej

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"etirement morphique"

[soloviev]

Dear colleagues,

who knows how to translate into English "etirement morphique"
and "prolixe" used by Penon, Kachour ("Cahiers de topologie...")?

All the best

Sergei Soloviev


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diagrams in computer algebra

[Sergei SOLOVIEV]

Question to the list:

I am very much interested, does there exist any practically usable
system for the "computer algebra style" work with diagrams. I.e. not
just write down the diagrams (like XyPic) but for example for
diagram chasing,
verification of commutativity in any interesting classes of categories
with structure etc. As I remember there were some systems permitting
computation of some limits in very limited classes of categories,
also some heavy attempts of formalisation in some general
purpose proof assistants
but I do not know about anything else.

Regards to all

Sergei Soloviev


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Re: diagrams in computer algebra

[Michael J Healy]

Sergei,

My colleagues and I have been looking for something like this for a project.  We need to be able to specify small categories as the completions of finite graphs we are given, extend these by specifying commutative diagrams, pullbacks, etc, of interest, then define functors generated from graph homomorphisms, and take colimits of diagrams in Cat, etc etc.  We haven't found anything that does all this.  So, we're programming it in Haskell---one of our grad students knows the language.   We'll be happy to share our experience and will probably make the code available.  It's a work in progress.

Best regards,
Mike Healy

On Oct 2, 2011, at 3:10 PM, Sergei SOLOVIEV wrote:

> Question to the list:
> 
> I am very much interested, does there exist any practically usable
> system for the "computer algebra style" work with diagrams. I.e. not
> just write down the diagrams (like XyPic) but for example for
> diagram chasing,
> verification of commutativity in any interesting classes of categories
> with structure etc. As I remember there were some systems permitting
> computation of some limits in very limited classes of categories,
> also some heavy attempts of formalisation in some general
> purpose proof assistants
> but I do not know about anything else.
> 
> Regards to all
> 
> Sergei Soloviev
> 


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Re: diagrams in computer algebra

[Eduardo J. Dubuc]

I paste and copy some old posting to this list that I have kept. It
seems to me that they are relevant for your discussion.

e.d.

> I'd like to announce a selection of Web-based category theory
> demonstrations that I've put up at
> http://www.j-paine.org/cgi-bin/webcats/webcats.php . The page contains a
> number of buttons such as "generate and demonstrate an equaliser" and
> "generate and demonstrate a limit": clicking on one will generate
> an example of the construct in the category of finite sets, and display it
> as a listing of its objects and arrows, and as a diagram. For limits and
> colimits, the demos generate a small random graph, convert it to a
> diagram, then compute and display its limit or colimit.
>
> Comments would be very welcome. The demos are a bit of an experiment: I
> had some categorical algorithms lying around from other work, and thought
> it would be interesting to connect them to the Web.


> "If you could commission a computer demonstration of any categorical idea,
> what would you ask for? Could such demonstrations have helped you, or your
> students, learn tricky ideas? And, would you be willing to share the
> visualisations and metaphors that you have devised to explain these ideas
> to yourself or others?"
>
> We've started a thread on this topic at the n-Category Cafe',
> http://golem.ph.utexas.edu/category/2009/04/graphical_category_theory_demo.html
> . Anybody interested in using computers to demonstrate concepts from
> category theory, do please have a look there. I've just added an
> explanation of the techniques available for delivering demonstrations over
> the Web, and I mention one - the 3D modelling environment called Alice -
> that I think will eventually be brilliant for animating category theory,
> and many other topics besides.


> This is to tell, or remind, readers about the Web-based interactive
> category-theory demonstrations I have on my site. Perhaps of interest to
> new students now an academic year is starting. They're at
> http://www.j-paine.org/cgi-bin/webcats/webcats.php . After some preamble,
> this page contains a form divided into sections. Each section generates a
> particular construct in the category of finite sets: e.g. a colimit,
> equaliser, or initial object. You can input sets and arrows, or let the
> demo choose its own. The output includes a diagram, and text explaining
> it.
>
> Cheers,
>
> Jocelyn Paine
> http://www.j-paine.org
> +44 (0)7768 534 091
>
> Jocelyn's Cartoons:
> http://www.j-paine.org/blog/jocelyns_cartoons/



On 03/10/11 15:02, Michael J Healy wrote:
> Sergei,
>
> My colleagues and I have been looking for something like this for a project.  We need to be able to specify small categories as the completions of finite graphs we are given, extend these by specifying commutative diagrams, pullbacks, etc, of interest, then define functors generated from graph homomorphisms, and take colimits of diagrams in Cat, etc etc.  We haven't found anything that does all this.  So, we're programming it in Haskell---one of our grad students knows the language.   We'll be happy to share our experience and will probably make the code available.  It's a work in progress.
>
> Best regards,
> Mike Healy
>

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Re: diagrams in computer algebra

[Michael J Healy]

Vasili,

Yes, I mean small.  All but FinSet (an appropriate version of it, anyway---certain functors into which are models, i.e.data respositories) are generated from finite graphs, through categorical completion elaborated with specified commutative diagrams, pullbacks, etc.

Regards,
Mike

On Oct 8, 2011, at 3:55 PM, Vasili I. Galchin wrote:

> Hi Mike,
> 
>     Do you really mean small category or do you mean finite category?
> If small category(potentially(<<< no pun intended) infinite), then I
> guess using Haskell is the right choice do to its lazy evaluation
> feature.
> 
> Regards,
> 
> Vasili
> 
> 
> On Mon, Oct 3, 2011 at 1:02 PM, Michael J Healy <mjhealy@ece.unm.edu> wrote:
>> Sergei,
>> 
>> My colleagues and I have been looking for something like this for a project.  We need to be able to specify small categories as the completions of finite graphs we are given, extend these by specifying commutative diagrams, pullbacks, etc, of interest, then define functors generated from graph homomorphisms, and take colimits of diagrams in Cat, etc etc.  We haven't found anything that does all this.  So, we're programming it in Haskell---one of our grad students knows the language.    We'll be happy to share our experience and will probably make the code available.  It's a work in progress.
>> 
>> Best regards,
>> Mike Healy
>> 

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Re: diagrams in computer algebra

[Michael J Healy]

Vasili,

Certainly---we'll be happy to do this for anybody who is interested.  Looks like I need to start a list.

I might mention again that we're leaning heavily on the work of Johnson and Rosebrugh.  Just for the record, though, any problems with anything we generate are strictly our own doing.  We're having to translate  adjunctions into software methods for generating the functors we need, and then the colimit theorem into software so we can generate ontologies as colimits in Cat.

Regards again,
Mike

On Oct 8, 2011, at 3:17 PM, Vasili I. Galchin wrote:

> Hi Mike and Sergei,
> 
>    I have also thought doing exactly this in Haskell but for dealing
> with sheafs. Is there any way Mike that I could monitor the progress
> of your effort assuming you will make source available as you said
> perhaps below?
> 
> Vasili
> 
> On Mon, Oct 3, 2011 at 1:02 PM, Michael J Healy <mjhealy@ece.unm.edu> wrote:
>> 
>> Sergei,
>> 
>> My colleagues and I have been looking for something like this for a project.  We need to be able to specify small categories as the completions of finite graphs we are given, extend these by specifying commutative diagrams, pullbacks, etc, of interest, then define functors generated from graph homomorphisms, and take colimits of diagrams in Cat, etc etc.  We haven't found anything that does all this.  So, we're programming it in Haskell---one of our grad students knows the language.    We'll be happy to share our experience and will probably make the code available.  It's a work in progress.
>> 
>> Best regards,
>> Mike Healy
>> 

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