April 1994 - Categories - categories.org.au

Cubical monads
by Marco Grandis 28 Apr '94

28 Apr '94

The following preprint is available "Cubical homotopical algebra and cochain algebras" Marco Grandis. Abstract. Basic homotopical algebra is developed in a setting consisting of a cubical monad (*), i.e. a cylinder endofunctor I, equipped with connections g-, g+: I^2 -> I, and - possibly - with symmetries extending the reversion r: I -> I and the interchange s: I^2 -> I^2 of the standard topological case. Our study is mostly concerned with the Puppe sequence of a map f and its comparison with the sequence of iterated homotopy cokernels of f. As an application, the homotopy structure of cochain algebras is studied in the present frame, through the cubical co-monad of the path functor P and the left adjoint cylinder functor I. This paper is a sequel of (*) "Cubical monads and their symmetries", whose abstract appeared in "CATEGORIES", Mon, 4 Oct 1993. Regards, Marco Grandis Dipartimento di Matematica, Universita di Genova, Via L.B. Alberti 4, I - 16132 Genova, Italia e-mail: grandis(a)dima.unige.it

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by France.Dacar＠ijs.si 28 Apr '94

28 Apr '94

> Date: Tue, 26 Apr 94 13:44:46 PDT > From: "Scott L. Burson" <gyro(a)zeta-soft.com> > > > Date: Tue, 26 Apr 94 09:19:01 EDT > From: Peter Freyd <pjf(a)saul.cis.upenn.edu> > > It seems that somewhere between 284 and 305 Diocletian proclaimed: > > Artem geometriae discere atque exercere publice interest, > ars autem mathematica damnabilis interdicta est omnino. > > Does that say what I think it does? Can someone supply an > authoritative translation? > > Yep, it means just what it looks like: > > To learn and practice the art of geometry is publicly of interest, but the > damnable mathematical art is proscribed altogether. > > -- Scott Appearances might be misleading. What precisely were the meanings of "geometria" and "mathematica" in Diocletian times? I suspect that "geometria" encompassed what we would call the whole of mathematics as known then, while "mathematica" was a branch of sorcery (benign as well as malign, ie. both white and black) that involved much high-fallutin' twiddling of symbols, mysterious diagrams and geometric figures, and such -- thus a kind of formalized abstract magick. (Some recent literature on uses of category theory in computer science in general and in AI in particular would be basic manuals, as it were, for those mathematically inclined sorcerers. They could use them quite effectively even if they would not understand what it's all about, since they could always pretend they do, and they were masters of illusionism.) Can somebody tell whether this is a good enough guess? ------------------------------------------------------------------------ France Dacar Email: france.dacar(a)ijs.si Computer Science Department Phone: +386 61 1-259-199 / 768 Jozef Stefan Institute Fax: +386 61 1-258-058 Jamova 39, 61000 Ljubljana, Slovenia

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The last of Diocletian
by Gavin Wraith 28 Apr '94

28 Apr '94

A propos the Diocletian quote, I support Max Kelly and John Isbell : I do not like to jump in with both feet without seeing the locus and its context, but I would say that the sense is "Surveying is generally acknowledged to be a social benefit but astrology is everywhere taken to be pernicious" "Geometria" at this time would have military overtones - design of fortifications, disposal of forces over terrain etc. "Mathematicus" almost certainly means "astrologer" or "soothsayer". Astrology was not respectable, being seen as a vulgar oriental import. Two centuries earlier, Juvenal in his sixth satire (line 562) says "Nemo mathematicus genium indemnatus habebit" (No astrologer gets a reputation until he is convicted) a line which the Diocletian quote could well be referring back to. Sorry if this is a disappointment - but a nice quote anyhow! -- Gavin Wraith ++++++++++++++++++++++++++++++++++++++

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by MTHISBEL＠ubvms.cc.buffalo.edu 28 Apr '94

28 Apr '94

The three translations I have seen here are very nice except for the meaning. *Geometry* means pretty much #mathematics#. *Mathematics* is #divination#. John

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by Peter Freyd 27 Apr '94

27 Apr '94

The mysterious proclamation of Diocletian, Artem geometriae discere atque exercere publice interest, ars autem mathematica damnabilis interdicta est omnino. (To learn and -- especially -- to practice the art of geometry is in the public interest but the damnable mathematical art is proscribed altogether.) was found as a citation for the once common use of the word "mathematicus" to mean "wizard". Another citation is from England (from Shirley's 1642 comedy "The Sisters"): Giovanni: Master Steward, yonder are the rarest fellows! In such fantastical habits too; they call themselves mathematicians. Steward: What do they come for? Giovanni: To offer their service to my Lady and tell fortunes. ... Antonio: Her house is open for these mountebanks, Cheaters, and tumblers, that can foist and flatter My lady Gewgaw... What are you, sir? Strozzo: One of the mathematicians, noble Signior. Antonio: Mathematicians! mongrel, How durst thou take that learned name upon thee? You are one of those knaves that stroll the country, And live by picking worms out of fools' fingers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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More on geometriae/mathematica
by Max Kelly 27 Apr '94

27 Apr '94

The Latin of Peter Freyd's quotation from Diocletian may be translated: To teach and exercise the art of geometry is in the public interest; the damnable art of mathematics, however, is forbidden to every man. The point, surely, is that in those days "mathematics" meant "astrology". See Augustine, in The City of God, if I recall correctly. In the twelfth century "Play of Daniel", as performed at Beauvais, when the writing on the wall appears, the king says "Vocate mathematicos" - "call the soothsayers (or astrologers). Regards, Max Kelly. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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by Walter 27 Apr '94

27 Apr '94

My translation would be: "To learn and (especially) to practice the art of geometry is generally of importance (or: is in the public interest) , the condemnable art of mathematics, however, is altogether forbidden." Walter Tholen.

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by Moez Alimohamed 27 Apr '94

27 Apr '94

The translation of Diocletian contained in the previous posting is due to the mathematician and Latin scholar Raymond Thomas (rlt(a)cunyvms1.gc.cuny.edu) ++++++++++++++++++++++++++++++++++++++++++++++++++++

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by Moez Alimohamed 27 Apr '94

27 Apr '94

Peter Freyd's writes: It seems that somewhere between 284 and 305 Diocletian proclaimed: Artem geometriae discere atque exercere publice interest, ars autem mathematica damnabilis interdicta est omnino. Does that say what I think it does? Can someone supply an authoritative translation? ------------------------------- In translation, this reads: Learning the art of geometry and furthermore, using it on behalf of the state is of interest - nevertheless, mathematics is altogether worthy of condemnation and is forbidden. Very strongly worded.

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What is a "principal fibration" ?
by Thomas Streicher 27 Apr '94

27 Apr '94

I have heard that in the Grothendieck school there has developed a notion of "principal fibration" which should generalize the notion of principal bundle ! Please, could anyone give me a reference ! Thomas Streicher

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