July 2017 - Categories - categories.org.au

Do there exist nontrivial locally bounded geometric morphisms and/or locally (pre)sheaf toposes?
by Mamuka Jibladze 02 Aug '17

02 Aug '17

Recently I posted this question https://mathoverflow.net/q/277582/41291 to mathoverflow and now it occurred to me that most likely I can get a quick answer here. Are there geometric morphisms f: YY -> XX which are (1) locally but not globally bounded, or (2) locally but not globally presheaf, or (3) as in (2) and bounded? In more detail, I mean this: there must be an object X in XX with global support (X->1 epic) such that the pullback f/X: YY/f^*(X) -> XX/X is (1) bounded, while f is not bounded, or (2) equivalent over XX/X to the topos (XX/X)^{CC^op} of internal presheaves on some internal category CC of XX/X, while YY is not equivalent to any such over XX, or (3) same as (2) and in addition f bounded. Can any of these happen? Hoping, Mamuka [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Postdoctoral Position
by Mislove, Michael W 26 Jul '17

26 Jul '17

Dear All, Below is an announcement of a postdoctoral research position starting October 1 and running through November 30, 2018. I encourage all interested parties to apply. Thanks, Mike Mislove =============================================== Michael Mislove Phone: +1 504 865-5803 <tel:%2B1%20504%20865-5803> Professor and Chair FAX: +1 504 865-5063 <tel:%2B1%20504%20865-5063> Department of Computer Science Tulane University URL: http://www.cs.tulane.edu/~mwm <http://www.cs.tulane.edu/~mwm> New Orleans, LA 70118 USA =============================================== Postdoctoral Researcher in Semantics and Tools for Quantum Programming Languages Applications are invited for a postdoctoral position beginning October 1, 2017 and running through November 30, 2018. The position is in the Department of Computer Science at Tulane University, and will be under the supervision of Professor Michael Mislove. The successful applicant will work on a project entitled "Semantics, Formal Reasoning, and Tool Support for Quantum Programming”. The project involves designing high-level semantic models and tools to support quantum functional programming languages. A prototype language is Proto-Quipper, which has been under development (http://www.mathstat.dal.ca/~selinger/quipper/ <http://www.mathstat.dal.ca/~selinger/quipper/> ). This language uses the circuit model for quantum computation, and envisions languages that support quantum computation under classical control. The overall aim is to design type-safe functional programming languages for quantum computing. The project also involves developing the meta-theory (including categorical semantics) of such languages, and eventually to formalize some of the meta-theory in a proof assistant. The focus of the Tulane work is modeling recursion in such languages, which requires developing quantum domain theory, but interactions with other aspects of the project are expected. Familiarity with programming language design, and / or semantics is a prerequisite of the position. The latter includes categorical semantics and domain theory. A good knowledge of category theory is also a prerequisite. Of course, familiarity with quantum computing is helpful. Additional components of the project will address issues around quantum information such as non-locality and contextuality, adapting proof assistants (Coq, Agda, Lean, etc) to develop automated verification for quantum programming languages. To apply for one of this position, direct your browser to the link: https://apply.interfolio.com/41053 <https://apply.interfolio.com/41053> Funding for the project comes from the DOD and the U.S. Air Force Office of Scientific Research. ######################################################################## [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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What is a contractible simplicial object?
by Michael Barr 21 Jul '17

21 Jul '17

A few weeks ago, I asked on this group how one defines contractibility of simplicial objects. Later I tried Math Overflow and discovered that the question had been asked but not satisfactorily answered several years ago. Below is a \TeX file that gives an entirely satisfactory answer. \def\dot{{_\bullet}} \def\to{\mathop{\longrightarrow}\limits} For an augmented simplicial set, the answer is clear. Given $X_\dot\to X_{-1}$, a contraction is a sequence of maps $t=t_n:X_n\to X_{n+1}$ for all $n\ge -1$ such that $d^0t=1$, $d^it=td^{i-1}$ for $i>0$, $s^0t=tt$, and $s^it=ts^{i-1}$ for $i>0$. These are the equations a degeneracy $s^{-1}$ would satisfy. Suppose $X\dot$ is a simplicial object in a category with split idempotents and there are maps $t=t_n:X_n\to X_{n+1}$ for all $n\ge0$ satisfying all the above equations as far as they are defined. We have, in $X_0$ $$d^1td^1t=d^1d^2tt=d^1d^1tt=d^1td^0t=d^1t$$ so that $d^1t$ is idempotent. If we factor it as $X_0\to^{d^0}X_{-1} \to^{t_{-1}}X_0$ we nearly have a contractible augmented simplicial object. There two things to verify: first that $d_0^0d_1^0=d_0^0d_1^1$ and second that $s_0^0t_{-1}=t_0t_{-1}$. I have used the normally omitted lower indices to clarify that the low dimension indentities follow from the higher ones. For the first, $$t_{-1}d_1^0d_2^0=d_0^1t_0d_1^0=d_1^1d_2^1t_2=d_1^1d_2^2t_1= d_1^1t_0d^1 =t_{-1}d_0^0d_1^1$$ and $t_{-1}$ being a (split) monic can be canceled to give the result. For the second, $$s_0^0t_{-1}d_0^0=s_0^0d_1^1t_0=d_2^2s_1^0t_0=d_2^2t_1t_0= t_0d_1^1t_0=t_0t_{-1}d_0^0$$ and $d_0^0$ being a (split) epic can be canceled to give the result. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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n-types and n-fold structures
by Paoli, Simona (Dr.) 20 Jul '17

20 Jul '17

Dear All, With regard to the recent postings about modelling general n-types and n-fold groupoids, I would like to point out the following. A proof of the modelling of general n-types (not only path-connected ones) using n-fold structures is given in the following research monograph, which I recently posted on arVix: S.Paoli, Segal-type models of higher categories, arXiv.1707.01868 The structure used there, called groupoidal weakly globular n-fold categories, is a subcategory of n-fold categories in which certain sub-structures are groupoidal, and satisfying several other conditions. This result is a proof of the homotopy hypothesis for a new model of weak n-categories, called weakly globular n-fold categories, which is proved to be suitably equivalent to the Tamsamani-Simpson model. In the above work there is also a proof that it is possible to use as fundamental functor from spaces to groupoidal weakly globular n-fold categories the construction found in the following paper D. Blanc, S. Paoli, Segal-type algebraic models of n-types, Algebraic and Geometric Topology, 14 3419–3491. This construction produces a functor from spaces to a subcategory of n-fold groupoids. Best wishes, Simona. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Ph.D. and post-doctoral positions in Como
by Olivia Caramello 19 Jul '17

19 Jul '17

Dear All, Applications are invited for Ph.D. studentships and post-doctoral fellowships at the Department of Science and High Technology of the University of Insubria in Como, Italy. We particularly welcome candidates who wish to work in areas represented within the Department, including logic and topos theory. The deadlines are the 1st of September 2017 for the Ph.D. studentships and the 8th of September 2017 for the post-doctoral fellowships. More details are available at the following pages: http://www4.uninsubria.it/on-line/home/naviga-per-tema/didattica/post-laurea m/dottorati-di-ricerca/articolo14089.html http://www4.uninsubria.it/on-line/home/naviga-per-tema/servizi/servizi-per-l a-ricerca/opportunita-di-ricerca-presso-luniversita-degli-studi-dellinsubria /assegni-di-ricerca-senior.html All interested applicants are welcome to send requests for further information or informal enquiries to their desired prospective supervisor. Please feel free to diffuse this information to any students or researchers you think may be interested. Best regards, Olivia Caramello [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Fully funded PhD studentship in Automated Verification through Coalgebras at the University of Southampton
by Corina Cirstea 19 Jul '17

19 Jul '17

PhD Studentship: Automated Verification through Coalgebras ********************************************************************** University of Southampton - Faculty of Physical Sciences and Engineering Qualification type: PhD Location: Southampton, UK Funding for: UK students, EU students Funding amount: EU/UK fees and stipend in line with EPSRC rates Hours: Full Time Deadline for applications: 4th August 2017 PhD Supervisor: Corina Cirstea Project Description: As part of this project, you will work on developing new foundations for quantitative verification, grounded in the theory of coalgebras. Coalgebras are mathematical structures suited for modelling a large variety of state-based, dynamical systems, with features such as non-deterministic, stochastic, or weighted behaviour being uniformly captured in the models. Coalgebras come with modal and temporal logics that expressively support reasoning about the temporal behaviour of dynamical systems, both qualitatively and quantitatively. Owing to the diversity of semantic models that can be captured, coalgebra is the ideal framework for modelling and reasoning about heterogeneous systems whose components have different quantitative verification concerns. During your PhD, you will help build a synergy between the fields of coalgebraic methods and automated verification, and will publish at internationally leading conferences and journals. Your project will contribute to the development of a new research area of coalgebraic model checking, aimed on the one hand at unifying existing quantitative model checking techniques, and on the other at extending the applicability of such techniques to complex systems, whose modelling requires a variety of underlying models. Application areas for such techniques include, but are not limited to: cloud computing, with a need for analyzing the efficiency of resource-management operations; industrial control systems, with a need for energy-aware controller synthesis; and cyber security, with a need for resource-aware reasoning about system-attacker interactions. More details about the project can be found here: http://www.jobs.ac.uk/job/BBD448/phd-studentship-automated-verification-thr… Candidate profile: • good undergraduate degree in mathematics or computer science (required), • interest in theoretical computer science/logic/automated verification (required), • good programming skills (desirable). For further details about the project, please contact Dr Corina Cirstea (cc2(a)ecs.soton.ac.uk) Application process: To apply, please email your CV, transcript of your degree and a brief statement of research interests to cc2(a)ecs.soton.ac.uk. ---------------------------------------------- Dr Corina Cirstea Electronics and Computer Science University of Southampton Southampton, SO17 1BJ United Kingdom Tel: +44 (0)23 8059 3625 ---------------------------------------------- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Homotopy hypothesis for contractible operad definitions of weak n-categories
by Timothy Porter 16 Jul '17

16 Jul '17

Ronnie, The point about pointed spaces is perhaps not really that difficult to get around at least partially. Using either the Bullejos-Cegarra-Duskin method or my approach the formulae for simplicial groups extends more or less to simplicially enriched groupoids and thus to all homotopy types. This works equally well for your point in the second paragraph, but the obscure part is the following: I will look back at the examples in arXiv:0903.2627v2, but I remember discussing this with you and one of the differences is that in a simplicially enriched groupoid, G, one has a simplicial groupoid with constant object of objects. What this means for translating the simplicial composition in G into structure in the corresponding cat^n objects is where interesting things start happening so writing L for Loday's construction one wants to use a composition G(x,y)xG(y,z)-> G(x,z) for give something at the level of LG(x,y)?LG(y,z)->LG(x,z), where ? is some construction not yet defined. This should be possible to analyse as the multiplication in a simplicial group leads to structures in the corresponding cat^n group and back again, so probably one could look at that as G(x,x)\times G(x,x)\to G(x,x) to see what is happening??? This is getting a bit off-topic for the original messages so I will stop, but would welcome any ideas either on a separate thread or via MathOverflow perhaps. Tim On 15 July 2017 at 21:59, RONALD BROWN <ronnie.profbrown(a)btinternet.com> wrote: > Dear All, > > Loday's model is for pointed spaces, and Grothendieck was critical of this > in a letter to me in 1983, of which I have quoted part in the Indag Paper > on my preprint page. I did not worry about this in the 1980s since the > immediate consequences were quite novel. For example, Ellis and Steiner > solved the old problem of the critical group for (n+1)-ads, and the > nonabelian tensor product of groups has been well developed by group > theorists (see www.groupoids.org.uk/nonabtens.html) > > What has not been looked at is an input of crossed modules over groupoids, > instead of over groups, and considering first the work of Ellis-Steiner in > that light. (crossed n-cubes of groupoids?) > > We know from examples that strict 2-fold groupoids are more complicated > than homotopy 2-types, see my preprint arXiv:0903.2627v2; and the van > Kampen theorem with Loday has not so far been given a version with many > base points, unlike the version in the book Nonabelian Algebraic Topology. > > The philosophy given in the Indag Paper has relatively recently been put > in this form, and so no part of it was discussed with Grothendieck, except > the idea that n-fold groupoids model homotopy n-types, which, as said > above, is not quite correct, though he thought it "absolutely beautiful". > At that time, 1985, he was starting to write "Recollte et Semaille", a > task which seemed to lead him away from mathematics. > > The work with Loday shows in many explicit examples how low dimensional > identifications in topology can give rise to high dimensional homotopy > invariants, and there are explicit and precise calculations using the > higher van Kampen theorems. Such calculation is not the only aim, but it > does give a useful test. > > Best > > Ronnie > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Homotopy hypothesis for contractible operad definitions of weak n-categories
by Jamie Vicary 15 Jul '17

15 Jul '17

Hi, Batanin, Leinster and other have presented related definitions of weak n-groupoid in terms of contractible globular operads. I personally find these definitions of "contractible n-groupoids" extremely beautiful. I am interested to learn what evidence we have that the homotopy hypothesis might be true for (at least one of) these definitions. Some good evidence is provided by Peter LeFanu Lumsdaine's [1] proof that a homotopy type gives rise to an infinity-groupoid in the sense of Leinster. There is other work along similar lines. But, as far as I am aware, it remains possible that contractible n-groupoids might in general be weaker structures than homotopy n-types. A fun way to investigate this would be to verify small instances of phenomena associated to the periodic table in contractible n-groupoids. For example, Christoph Dorn has shown me a proof that the Eckmann-Hilton argument holds in a Leinster 2-category; that is, for an object X, and for 2-morphisms f,g:id[X]-->id[X], we have f.g=g.f, thereby establishing one of the first phenomena predicted by the periodic table. Have any higher phenomena from the periodic table been verified? Or, is there other evidence that contractible n-groupoids behave "homotopically" in general? Best wishes, Jamie [1] http://peterlefanulumsdaine.com/research/Lumsdaine-Weak-omega-cats-from-ITT… [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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some new preprints/presentation
by RONALD BROWN 15 Jul '17

15 Jul '17

"Modelling and Computing Homotopy Types: I" (now published in Indagationes Math) 25 pages (Sections 1,2 of this paper outline a philosophy and methodology for using strict higher groupoids in algebraic topology) "Higher dimensional modelling and rewriting in algebraic topology" 7 pages "Homotopical Excision: (work to be done!)" (beamer presentation: 19 pages) (This last begins a new look at the excision theorem of the 1987 paper with Loday "Homotopical Excision and Hurewicz Theorems for n-cubes of spaces". ) These are available at http://groupoids.org.uk/brownpr.html Best wishes Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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CT Oktoberfest in Pittsburgh 10/28-29
by Steve Awodey 13 Jul '17

13 Jul '17

Dear Categories, This is the first announcement of a Category Theory Oktoberfest to be held the weekend of October 28-29, 2017, in Pittsburgh at Carnegie Mellon. More information will follow later in the summer, but please do pencil in the dates. Looking forward to seeing you in Pittsburgh! Steve [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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