Wednesday, September 17, 2014

Giuseppe Peano

On Saturday I  bought for 2 euros  at the mercatino of Lavello a little book by Giuseppe Peano of numerical tables. It has an interesting preface by Peano.

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Tuesday, September 16, 2014

Ricordo di Aurelio Carboni, Matematico e Filosofo

I have just received a volume of "Il Protagora" in which there is a section devoted to Aurelio's memory.
There are four articles all in
Il Protagora, Volume XL, July-December 2013, sesta serie, n. 20
The articles are
Fabio Minazzi, Un ricordo di Aurelio Carboni, pp. 489-494
F. William Lawvere, Farewell to Aurelio, pp. 495-498
R.F.C. Walters, Working with Aurelio - Tangled Lives, pp. 499-504
George Janelidze, An open letter to Aurelio Carboni and all Mathematicians who remember him, pp. 505-513


Prepublication versions of three of the papers are at the Como Category Archive.

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Sunday, September 14, 2014

Lex total categories and Grothendieck toposes IV

Previous post in this series; next post.

The aim of this post is to complete the proof that well-powered lex total categories are elementary toposes, by proving that they have subobject classifiers.

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Thursday, September 04, 2014

Span(Graph) III: Circuits with feedback

Previous post in this series; next post in the series.

The two main operations in $\bf Circ$ are composition and parallel as for straight-line circuits, and there are a variety of constants.

Today we describe the operation of composition of circuits with feedback in $\bf Circ$.

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Wednesday, September 03, 2014

Lex total categories and Grothendieck toposes III


Previous post in this series; next post in this series.

Today I want to give an application of the adjoint functor theorem for totally complete categories, namely to show that lex total categories are cartesian closed. The proof should be a generalization of the proof that locales are cartesian closed so we should look at that before attempting the case of lex total categories. We will be repeating some of the discussion of the first post in this series.

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Friday, August 15, 2014

Span(Graph) II: Circuits with feedback

Previous Span(Graph) post; next Span(Graph) post.

As an introduction to ${\bf Span(Graph)}$ I will describe how to extend the category of straight-line circuits to allow circuits with state and feedback. But before doing that I would like to point out a couple of things about straight-line circuits.

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Wednesday, August 13, 2014

Lex total categories and Grothendieck toposes II

Next post in this series.

I said in the first post in this series that totally complete categories have a strong adjoint functor theorem.

Here is the adjoint functor theorem: if $A$ and $B$ are locally small categories, $A$ is totally cocomplete and $F: A\to B$ is a functor which preserves colimits of discrete fibrations with small fibres then $F$ has a right adjoint.

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