Re: Subobject Classifier Algorithm

["Eduardo J. Dubuc" <edubuc@dm.uba.ar>]

I copy an old post in the list that may be of interest to the present matter
and that I had saved by curiosity but not acted upon afterwards.

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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/
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greetings  e.d.

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

Ellis D. Cooper wrote:
> P.20 of Prof. Taylor's book briefly recounts the history of
> "function" as a (rigorously formulated) expression for numerical
> calculation using arithmetic and transcendental operations. More
> generally, Cox et al in "Ideals, Varieties, and Algorithms" define
> "algorithm" as a (rigorously formulated) set of instructions for
> manipulating input expressions resulting in output expressions.
> Algorithms may be presented in "pseudocode" as a prelude to
> implementation in a particular computer programming language such as
> Maple, or Haskell.
>
> Mac Lane-Moerdijk define an elementary (Lawvere-Tierney) topos to be
> a category with finite limits, finite colimits, exponentials, and a
> subobject classifier. So to prove a category is a topos it is
> necessary to prove that it has a subobject classifier.
>
> My query was stimulated by Lawvere-Schanuel in "Conceptual
> Mathematics" pp.340-341 proof that the category of directed graphs
> has a subobject classifier. They give a finite list of the
> possibilities for an element of a graph (dot or arrow) to belong to a
> subgraph. It seems to me such a list could be generated by an
> algorithm. Then there is a step explained by pictures leading to
> Omega(DirectedGraph). To me this hints at an algorithm too.
>
> Ellis D. Cooper
>
>

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