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. ________________________________________________________________________ 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/ _______________________________________________________________________ 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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