April 2014 - Categories - categories.org.au

Partial functor
by David Leduc 16 Mar '15

16 Mar '15

Hi, A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Question on factorization systems
by Michael Barr 05 May '14

05 May '14

First let me explain that our math dept email system has been down for ten days and there is no indication when it will be back, although our sysop has been working on it day and night. I will circulate an announcement when it is running again. Meantime, use this address. Is there one place that develops all the properties of factorization systems? We are especially interested in the non-strict case, that is in which the right factor needn't be epic, nor the left factor be monic, but the unique diagonal fill-in condition holds. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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ResearchGate
by Ieke Moerdijk 03 May '14

03 May '14

Dear colleagues, Every week, I receive an invitation "from one of you" to join ResearchGate, whatever this is. And some of you may receive invitations "from me". I have been ignoring these e-mails, but am getting more and more annoyed. There are some people out there building up a data base, which they will next sell to interested parties, such as commercial publishers. What they are doing is probably illegal, in the sense that, for example, they are claiming to write "on Joyal's behalf" in the case copied below. I believe we should do something about this. Do we have a legal expert in our midst? Are we going to claim our share when they sell? I'd be interested to hear your opinions. Ieke. -------- Original Message -------- Subject: Reminder: Joyal Andre and 1 other invited you to join ResearchGate Date: Thu, 17 Apr 2014 11:42:24 +0000 (UTC) From: Joyal Andre <no-reply(a)researchgate.net> To: i.moerdijk(a)math.ru.nl ResearchGate [profile_default_m.jpg] Your co-author Joyal Andre invited you to join their network on ResearchGate and confirm authorship of your publications. [profile_default_m.png] Ieke Moerdijk 86 Publications · 6.94 Impact Points · 876 Citations [smallpreview.png] Orbifolds, Sheaves and Groupoids K-Theory 06/1997 Uploaded by Dorette Pronk + 85 other publications Confirm authorship I am not this author 1 pending invitation, including: Anders Kock Anders Kock Invitation to confirm authorship of your publications Accept This message was sent to i.moerdijk(a)math.ru.nl. To make sure you receive our updates, add ResearchGate to your address book or safe list. See instructions Not interested in invitations to join ResearchGate? Click here. ResearchGate GmbH, Invalidenstr. 115, 10115 Berlin, Germany. See our Privacy Policy and Terms & Conditions. [footer.gif?uid=01086534fbdb055fe6&cn=n9x4sYWEgEAjwi_FLe28NzWeXrQ64ZJAhki28 KPFru1_HUpr_OcymwTQsCVzjngyd26TD-4KmQfXOfBfx0beNmMrhmsSYTcwhU_Lq0sP8EyImRxw IblQ9cgC3PHTzPlF] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Higher Structures along the Lower Rhine V
by Ieke Moerdijk 29 Apr '14

29 Apr '14

Dear colleagues, The next meeting of the joint Bonn - Nijmegen - Utrecht seminar, /Higher Structures along the Lower Rhine V,/ will take place in Nijmegen on June 19-20. Speakers are Breen, Harpaz, Kaufmann, Lambrechts, Sabatini, Schreiber and Sepe. Participation if free, but registration is necessary for the obvious logistical reasons. For further information, see http://www.ru.nl/math/research/vmconferences/2014-higher/ Ieke Moerdijk. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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history_preprint
by Eduardo J. Dubuc 28 Apr '14

28 Apr '14

The following preprint is avaiable: This Note describe my own recollection of the first 30 years of Category Theory, it is not the result of any historical investigation. The choice of concepts and its evaluation is my own, necessarily subjective. It follows a chronological line along the 21 references in the bibliography. It is a faithful version, based on 10 manuscript transparencies, of a conference given in October 2013 at the fourth ENHEM* congress in Cali, Colombia. As such, its content is bounded by time, and it does not include anything that was not said there. Text is in spanish. http://arxiv.org/pdf/1404.6240v1.pdf * Escuela Nacional de Historia y Educacion Matematica. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Answer to Validity of equations ...
by Eduardo J. Dubuc 28 Apr '14

28 Apr '14

The question has been clarified: Andreas Blass in Mathoverflow argues and come to the conclusion: "The moral of this story is that, when deducing geometric sequents from other geometric sequents, we lose no generality by assuming the internal axiom of choice (and, with it, the law of the excluded middle)." Phil Scott does the same in private mail and comes to the conclusion: "So any theorem in that language (geometric theory) true in Sets (or provable using classical logic and AC) is necessarily true in any Grothendieck topos." So, for example, if we want to prove that the validity of some equation follows from the validity of other equations in the real interval [0, 1] (equations in any geometric language interpretable in [0, 1]), then we can proceed by cases, for example, (x = 1) or (not x = 1) (which with an order relation interpretable as the usual order) amounts to (x = 1) or (x < 1). Even though that [0, 1] is not decidable. So this proof which is not valid intuitionistically will in practice be accepted as a proof in the internal language of any topos, since we accept the conclusion as true without requiring any other proof. I confess this comes as a surprise to me. Cheers, as Fred would say. e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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postdoc in Seville
by Fernando Muro 28 Apr '14

28 Apr '14

There will be very soon an open postdoc position at the Math Institute of the University of Seville (IMUS, http://www.imus.us.es) The contract is for full-year periods up to 4 years (it has to be renovated every year, but this is straightforward). The salary will be 32.000??? per year. There are no teaching duties, but it is possible to teach if the postdoc wishes. Candidates must have obtained their PhD not before 2004. The position is not attached to any specific area of mathematics. Category theorists are very welcome to apply. Potential candidates can contact me for further information. Please, share this with whoever you think that may be interested. Best, Fernando Muro -- Fernando Muro Universidad de Sevilla, Departamento de ??lgebra http://personal.us.es/fmuro [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Validity_of_equations_in_topos
by Eduardo J. Dubuc 27 Apr '14

27 Apr '14

I clarify my previous question with a simple example. It is also a simpler (and more general) question whose resolution resolves the previous question on interpreting a theory in another. QUESTION: If an equation (in an algebraic theory) can be proved to hold by any means (choice, excluded middle, etc) in any Set-Model, then it holds in any model in any topos. No need to do an intuitionistic proof ? For example if we use the excluded middle in any totally ordered poset, (this includes the real interval [0, 1]) Example: The reference is the book [CDM] ``Algebraic foundations of Many-valued Reasoning?? Kluver Academic Publishers (2000) Consider the theory of MV-algebras. By a standard choice argument we have: 1) Every MV-algebra is a subalgebra of a product of MV-chains with the pointwise structure. Corollary. Given any two terms p(x, y, ... z) and q(x, y, ... z), the equation p(x, y, \ldots z) = q(x, y, \ldots z) holds in every MV-algebra if and only if it holds in every MV-chain. (denote "@" for $\oplus$, and "*" for $\odot$ 2) x @ y < 1 ==> x * y = 0 holds in any MV-chain. (very easy, [CDM] 1.6.1 page 27). 3) $x @ 0 = x$ (axiom, [CDM] page 7), $1 @ x = 1$ (very easy, [CDM] page 9). 4) The equation (a): (x @ y) @ (x * y) = (x @ y) holds in any MV-algebra. proof: By 1) it is enough to prove it for MV-chains: case (x @ y) = 1: we have 1 @ (x * y) = 1 case (x @ y) < 1: we have (x @ y) @ 0 = (x @ y) (by 2) both cases hold by 3). 5) By the completeness theorem the equation (a) follows from the axioms of the theory, (intuitionistically or classically ?) thus the equation should hold in (the internal language) for any MV-algebra object in any Grothendieck topos ? 5) Also we can argue via the presheaf classifying topos of the theory of MV-algebras and come to the same conclusion ? Is 5) valid ? Notice that, besides the use of choice in 1), if we have already a chain, we do not need 1), but we are using an exclude middle in 4) to prove the equation (invalid for example in the MV-chain [0, 1], the real interval in the object of reals of the topos). Never the less, the equation holds in [0, 1] but the proof does not ? e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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We're back
by Michael Barr 26 Apr '14

26 Apr '14

I am happy to tell you that the dept. email is back as of this morning. So barr(a)math.mcgill.ca works. Although I will keep the gmail account, I will not monitor it regularly. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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New preprints
by Olivia Caramello 25 Apr '14

25 Apr '14

Dear All, The following preprints are available from the Mathematics ArXiv. O. Caramello, "Extensions of flat functors and theories of presheaf type" Abstract: We develop a general theory of extensions of flat functors along geometric morphisms of toposes, and apply it to the study of the class of theories whose classifying topos is equivalent to a presheaf topos. As a result, we obtain a characterization theorem providing necessary and sufficient semantic conditions for a theory to be of presheaf type. This theorem subsumes all the previous partial results obtained on the subject and has several corollaries which can be used in practice for testing whether a given theory is of presheaf type as well as for generating new examples of theories belonging to this class. Along the way, we establish a number of other results of independent interest, including developments about colimits in the context of indexed categories, expansions of geometric theories and methods for constructing theories classified by a given presheaf topos. http://arxiv.org/abs/1404.4610 (158 pages) O. Caramello and A. C. Russo, "The Morita-equivalence between MV-algebras and abelian l-groups with strong unit" Abstract: We show that the theory of MV-algebras is Morita-equivalent to that of abelian l-groups with strong unit. This generalizes the well-known equivalence between the categories of set-based models of the two theories established by D. Mundici in 1986, and allows to transfer properties and results across them by using the methods of topos theory. We discuss several applications, including a sheaf-theoretic version of Mundici's equivalence and a bijective correspondence between the geometric theory extensions of the two theories. http://arxiv.org/abs/1312.1272 (29 pages) Links to video-recordings of talks in French about these two papers are available at the address http://www.oliviacaramello.com/Videos/Videos.htm . Best regards, Olivia Caramello [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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