September 2021 - Categories - categories.org.au

Re: "First" use of 'Category theory' to describe our field
by David Roberts 24 Feb '24

24 Feb '24

Dear Ross, no, that's pretty good! It's mildly surprising that it took ~20 years for the name to 'stick', but maybe less so given that the field grew slowly to start. Thanks, David David Roberts Webpage: https://ncatlab.org/nlab/show/David+Roberts Blog: https://thehighergeometer.wordpress.com On Thu, 11 Jul 2019 at 08:21, Ross Street <ross.street(a)mq.edu.au> wrote: > > Dear David > > From memory, the Pure Mathematics Honours (USyd) course that Max Kelly taught in 1965 > was called ``Category Theory''. It concentrated on different kinds of morphisms > and factorizations in a category, and finished with adjoint functors. > Also John Gray's (UIllinois) 1968-69 graduate course had that name. > > From Eilenberg I heard that each person using categories should have their own category of expertise. > I told this to John Gray who said that was fine; the time had come for that category to be Cat. > > I would suggest that the first category theorists to think of themselves as such were Eilenberg's students at Columbia. > However, Mac Lane was definitely a category theorist. > > This is probably not the verifiable stuff you were seeking. > > Ross > > > On 10 Jul 2019, at 10:01 PM, David Roberts <droberts.65537(a)gmail.com> wrote: > > Hi all, > > the (idle) question is: when did the phrase 'category theory' catch on > for the field? Clearly it didn't leap from either of the heads of > Eilenberg or Mac Lane full-grown, since they used the phrase 'General > theory of natural equivalences'. There are the old 'Reports of the > Midwest Category Seminar' lecture notes (the first in 1967), which > hints that 'category theory' wasn't quite the name in use. > > Even more interesting: who was the first "category theorist", by that name? > > Answers referring to verifiable sources would be best. > > Thoughts? > > David > > > > David Roberts > Webpage: https://ncatlab.org/nlab/show/David+Roberts > Blog: https://thehighergeometer.wordpress.com > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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One-year postdoc position on category theory for lambda-calculus and proof theory, in the Paris area
by Noam Zeilberger 30 Sep '21

30 Sep '21

CAUTION: The Sender of this email is not from within Dalhousie. We (Gabriel Scherer and Noam Zeilberger) are looking for candidates for a one-year post-doc position, ideally starting in January 2022, at INRIA Saclay / École Polytechnique (in the Paris area, France), as part of the [Partout](https://team.inria.fr/partout/) research group. Broadly, the subject of the postdoc will be categorical perspectives on the lambda-calculus and proof theory. Potential topics of interest include, for example, (multi)categorical structures capturing notions of focusing, as usually formulated in sequent calculi for various logics, as well as fibrational structures capturing different notions of type inference for lambda-calculi. Interested applicants should contact us directly: Gabriel Scherer <gabriel.scherer(a)inria.fr> Noam Zeilberger <noam.zeilberger(a)lix.polytechnique.fr> please include a CV with your application. Best regards, [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Proof Theory Seminar, Autumn 2021
by anupamdotdas＠gmail.com 28 Sep '21

28 Sep '21

CAUTION: The Sender of this email is not from within Dalhousie. Dear all, We are pleased to announce that the Proof Theory Virtual Seminar series will restart this autumn on Wednesday 6 October with a talk by Professor Jeremy Avigad. A list of this season's speakers can be found below, as well as other useful information. The Proof Theory Virtual Seminar is an online seminar series that presents talks by leading researchers from all areas of proof theory. It was launched in October 2020 and typically has talks on the first and third Wednesdays of the month, alternating between morning (9am UTC) and afternoon (5pm UTC) to facilitate international participation. 6 October, 17:00 UTC. Jeremy Avigad (Carnegie Mellon University). Title: The conservativity of weak K??nig's lemma (a proof from the book). 20 October, 09:00 UTC. Fedor Pakhomov (Ghent University and Steklov Mathematical Institute) Title: Fast growing hierarchies, ordinal collapsing, and ???????-CA???. 3 November, 17:00 UTC. Wilfried Sieg (Carnegie Mellon University). Title: Proofs as objects. 17 November, 09:00 UTC. Alexis Saurin (Centre National de la Recherche Scientifique) Title: (TBA) 1 December. (No seminar due to Proof Society Winter School and Workshop, 29 November - 4 December). 15 December, 09:00 UTC. Alessio Guglielmi (University of Bath). Title: (TBA) Joining details, abstracts and other information can be found on the seminar website: https://www.proofsociety.org/proof-theory-seminar/ If you would like to receive reminders, you can sign up to our email list by joining the following google group (no google account required): https://groups.google.com/forum/#!forum/proof-theory-seminar/join Best wishes, Lev Beklemishev, Yong Cheng, Anupam Das, Anton Freund, Thomas Powell, Sam Sanders, Monika Seisenberger, Andrei Sipos, Henry Towsner [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Parts and telling apart
by Phillip-Jan van Zyl 28 Sep '21

28 Sep '21

CAUTION: The Sender of this email is not from within Dalhousie. Dear Posina I am certainly not able to answer all of your questions in a way to captures every subtlety therein, but I can offer the following insights: 1. Functions between sets are indeed prototypical in the creation of a category, but you can always regard a function simply as a relation. Then all the functions have inverse relations. If you want to argue about the "inside" of such relations, you can do this by chasing subobjects. Chasing subobjects seems to be one fruitful way of getting around the fact that not all functions are invertible. When arguing about sets you can chase subsets, but you can also chase partitions. 2. I am not sure what a coseparator is, but note that nothing stops you from studying functions 2 -> X in a category. You can classify partitions in Sets in this way, by selecting pairs of (possibly the same) elements in equivalence classes. If you select distinct elements, this means that you select instead elements in different equivalance classes. In the category of groups a homomorphism 2 -> G is forced to give you a torsion element of order 2. In other words, it selects such g in G that g * g = 1. 3. In so far as pointedness is a categorical concept, the category of sets has certain behaviour that is pointed. If you take any function 1 -> X, then this induces a image, i.e. the singleton {f(x)}. But if you induce instead a partition, you always induce the discrete partition: the partition where every point is in its own equivalence class. This is a way to study Set such that it mimics the attribute in groups where f(1) = 1, where 1 is the group identity, i.e. constant functions. You can instead study the category of pointed sets, or the dual category of the category of sets. Partitions are subobjects in the dual category of Sets. Note that I am not a regular (person) in category theory circles so my definitions and arguments are not guaranteed to be standard. Best regards Phillip-Jan van Zyl ‐‐‐‐‐‐‐ Original Message ‐‐‐‐‐‐‐ On Saturday, September 25th, 2021 at 4:41 AM, Posina Venkata Rayudu <posinavrayudu(a)gmail.com> wrote: > I just thought of adding that the conditions for the representability > > of a category C as a functor category B^A is given my Roos' theorem > > (kindly translated by Professor Lafforgue): > > Roos' Theorem, as stated on page 415 of SGA4 (volume 1), says that the > > three following conditions on a topos E are equivalent : > > i) The family of essential points of E is conservative. (NB : A point > > is called "essential" when its fiber functor not only has a right > > adjoint but also a left adjoint.) > > ii) The full sub-category of E consisting in objects which are > > connected - non empty and projective is generating. > > iii) E is a presheaf topos. > > At the end of the volume, there is a reference to three notes (in > > French) of Roos at "Comptes Rendus de l'Académie des Sciences", with > > the general title "Distributivity of colimits with respect to limits > > in toposes". This reference is to CR 259 (August and September 1964) : > > p. 969-972, 1605-1608 and 1801-1804. > > In the context of the mutual relationship between change/variation and > > unity/cohesion, I'd like to add: > > A monoid with a single constant (as a kind of variation) gives rise to > > a topos (of right actions of the monoid), i.e. idempotents (Ex. 5 in > > Conceptual Mathematics, p. 367; which is a quality type; Definition 1 > > in http://tac.mta.ca/tac/volumes/19/3/19-03.pdf) whose subobject > > classifier is not connected, and hence fails to satisfy one of the > > axioms of cohesion (Axiom 2 in > > http://tac.mta.ca/tac/reprints/articles/9/tr9.pdf) > > On the other hand, a monoid with two constants (another kind of > > variation) gives rise to a cohesive topos (reflexive graphs; > > Conceptual Mathematics, p. 367). > > With regard to UNIVERSAL MAPPING PROPERTY, it was also not discovered > > in its "simplest" (structureless) instantiation of initial set (as > > that which has exactly one function to any set); it required a more > > structured setting: > > Samuel, P. (1948) On universal mappings and free topological groups, > > Bull. A.M.S. 54: 591-598 > > (https://www.ams.org/journals/bull/1948-54-06/S0002-9904-1948-09052-8/S0002-…) > > Is it related to the difficulty of seeing function as a > > structure-preserving map, given that the structure preserved is > > structurelessness? > > Lastly, is the > > coseparator ~ subobject classifier > > (isomorphism, after discounting that the 'pointed' in pointed object > > definition of subobject classifier) valid in the discrete/constant > > subcategory of any topos (category with subobject classifier)? > > I eagerly look forward to your corrections. > > Thank you, > > posina > > On Fri, Sep 24, 2021 at 3:54 AM Posina Venkata Rayudu > > posinavrayudu(a)gmail.com wrote: > >> Dear All, >> >> I hope and pray you and your families are all well. >> >> If I may, as I was preparing to give a lecture on SUBOBJECT >> >> CLASSIFIER, as part of the Conceptual Mathematics seminar series at >> >> Poornaprajna Institute of Scientific Research (https://ppisr.res.in/) >> >> I started recollecting how much I loved the formula 2^|X| for the >> >> number of subsets of a set X, when I first learned it in high school >> >> (it was simple :-). I remember listing out all the eight subsets in >> >> the case of X = {a, b, c}. (Grothendieck's profound insight of >> >> defining subsets as 1-1 functions was definitely not part of my >> >> understanding of SUBSET.) And about learning that the number of >> >> functions from a domain set X to a codomain set Y is |Y|^|X|, and with >> >> |Y| = 2 we could clearly see that: >> >> the number of subsets of a set X = the number of functions from the >> >> set X to a two-element set 2 = 2^|X| >> >> It's not out of the realm of possibilities that I might have listed >> >> all the functions from X to 2, which, along with all the subsets of X, >> >> would have brought the 1-1 correspondence: >> >> parts of a set X >> ---------------- >> >> functions from X to 2 >> >> into a clear view. >> >> Be that as it may, what occluded SUBOBJECT CLASSIFIER from set >> >> theorists and mathematical logicians; even Grothendieck missed it, but >> >> in his characteristic kindness called it Lawvere element, upon >> >> Professor F. William Lawvere's definition of subobject classifier as >> >> part of his axiomatization of topos (please see p. 7, >> >> http://www.mat.uc.pt/~picado/lawvere/interview.pdf) >> >> This is not an isolated incident in science; it appears to be a >> >> pattern--wherein far-reaching constructions are not initially >> >> conceptualized/recognized in their simplest instantiation, which is >> >> where the figural salience of the concepts is clearly visible for all >> >> to see and use--in scientific practice. A similar case can be made >> >> about universal mapping properties (e.g. terminal set 1 = {*}) and >> >> about category theory itself. That sets and functions form a >> >> mathematical category is not easy to ignore, but category theory took >> >> birth in an inaccessible realm rather remote for sets and functions >> >> (thanks to the then prevalent practice of identifying a function with >> >> its graph ((a, f(a)); cf. Conceptual Mathematics, pp. 293-294). >> >> Be that as it may, subset [and its representability by maps to >> >> subobject classifier] is also related to the telling apart the figures >> >> constituting the inside of an object of a category (with subobject >> >> classifier or topos; see Sets for Mathematics, pp. 18-21). In the >> >> case of sets, with coseparator 2 = {*, } as the property type, there >> >> are enough properties to tell apart any two different 1-shaped figures >> >> (points/elements) in any set. Does subobject classifier (in every >> >> category with it/topos) always serve as the property type coadequate >> >> to tell apart figures in any object of the category/topos? What >> >> difference does the fact that subobject classifier is defined as a >> >> pointed object (map from the terminal object of the category to the >> >> subobject classifier, i.e., t: 1 --> 2, where t() = t; see Exercise 8 >> >> in Conceptual Mathematics, p. 337), which is not the case with >> >> coseparator, i.e. it is just a constant/discrete/abstract 2-element >> >> set. It seems a little odd that representing parts of an object >> >> amounts to telling apart all different figures constituting the >> >> object, simply going by the fact that PART (monomorphism) is a special >> >> type of FIGURE (a morphism A --> B is an A-shaped figure in B; >> >> Conceptual Mathematics, pp. 81-85; for additional context, among other >> >> mathematical clarifications, see >> >> https://cgasa.sbu.ac.ir/article_12425_b4ce2ab0ae3a843f00ff011b054f918b.pdf) >> >> Telling apart figures in a domain object also appears to be the job of >> >> epimorphisms (with subobject classifier of the category as codomain >> >> object). >> >> Is >> >> coseparator ~ subobject classifier >> >> (isomorphism; after discounting that the 'pointed' in pointed object >> >> definition of subobject classifier) specific to localic (as opposed to >> >> cohesive) toposes (cf. Sets for Mathematics, pp. 93-94)? >> >> Also from a pedagogical perspective, given that our everyday >> >> experience is that of categories of objects, all of which partake in >> >> the essence/theory T characterizing a category C (Sets for >> >> Mathematics, pp. 154-155), and since all objects of any category >> >> partake in the essence T of the category, the transformations between >> >> objects of the category necessarily preserve the essence, and hence >> >> are structure-respecting maps (see Conceptual Mathematics, pp. >> >> 149-151), which, in turn, are representable as natural transformations >> >> (ibid, p. 378). Equally accessible is the idea that mathematical >> >> objects, which are about everyday objects, are not unlike everyday >> >> objects in that they are also made up of figures of various basic >> >> shapes and their incidences (my attempt to introduce basic shapes and >> >> their incidence relations to designers didn't get far; >> >> https://zenodo.org/record/3924760#.YUzpptJBzZ4) And, then, in >> >> addition to cohesion/essence/theory characterizing a category (in a >> >> sense, we are limiting ourselves to presheaves or those categories C >> >> that can be represented as contravariant functors M: T^op --> S, i.e. >> >> as diagrams in the category S of sets), we can see that the natural >> >> transformation (not unlike the transformations that we encounter in >> >> our everyday life such as a water flowing downhill) respects the >> >> essence of the object of a category that is being transformed (which >> >> is T in the natural transformation of M: T^op --> S to N: T^op --> S). >> >> Equally importantly, we can also begin with a mode of variation/change >> >> and arrive at a category of objects (topos of right actions of a >> >> monoid [objectifying change/variation]; Conceptual Mathematics, pp. >> >> 360-361). This direction, i.e. objectification of change is >> >> particularly important, given that we are given change (the basic >> >> building block of our conscious experiences is contrast) and those >> >> changes (natural transformations), in respecting/preserving the way >> >> figures of various basic shapes stick together, make it possible to >> >> reconstruct objects (as functors) based on the given change/variation >> >> (but for natural transformations preserving the unity of objects >> >> transformed, by the time I reach Malabar Cafe for ginger tea, there is >> >> nothing stopping my leg being up in the clear skies of Bengaluru ;-) >> >> Is it inappropriate to claim that a universe of everyday experience is >> >> accounted for by the notion of: >> >> categories of objects >> >> along with the mutual determination of: >> >> change/variation <--> cohesion/unity >> >> and does so in a manner accessible to total beginners (here's an >> >> everyday objectification of change; https://youtu.be/r0kLC-pridI) >> >> Your time permitting, please correct any mistakes I might have made in >> >> my characterization of categories of objects. >> >> I eagerly look forward to your corrections. >> >> Thanking you, >> >> posina > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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SYCO 8 - Call for Papers
by Maaike Zwart 27 Sep '21

27 Sep '21

CAUTION: The Sender of this email is not from within Dalhousie. -------------------------------------------------------------- CALL FOR PAPERS EIGHTH SYMPOSIUM ON COMPOSITIONAL STRUCTURES (SYCO 8) Tallinn University of Technology, Estonia 13-14 December 2021 https://www.cl.cam.ac.uk/events/syco/8/ -------------------------------------------------------------- After a long break, SYCO is back! The Symposium on Compositional Structures (SYCO) is an interdisciplinary series of meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. Previous SYCO events have been held at University of Birmingham, University of Strathclyde, University of Oxford, Chapman University, and University of Leicester. We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering friendly discussion, disseminating new ideas, and spreading knowledge between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students. -------------------------------------------------------------- RESTART OF SYCO The main aim of SYCO is to bring the community together and exchange ideas across interdisciplinary fields. In-person contact facilitates this in a way that online meetings cannot. This meant that unfortunately, SYCO 7, the previous edition of SYCO, had to be cancelled. To restart the series, SYCO 8 will take the following into account: - Submissions that were accepted for SYCO 7 can be resubmitted for SYCO 8, and will be given priority over other submissions. These submissions will be treated as being deferred from SYCO 7, see submission instructions below. - Authors that had a submission accepted for SYCO can also submit new material. These submissions will also be given priority over other submissions, if accepted through peer-review. SYCO 8 will be mainly an in-person event, and we expect that most speakers will be physically present. However, we do support a hybrid format, so if for some reason you cannot be present in person but would still like to present your work at SYCO, please contact the programme chair, Maaike Zwart (maaike.annebeth(a)gmail.com) -------------------------------------------------------------- TOPICS OF INTEREST While no list of topics could be exhaustive, SYCO welcomes submissions with a compositional focus related to any of the following areas, in particular from the perspective of category theory: - logical methods in computer science, including classical and quantum programming, type theory, concurrency, natural language processing and machine learning; - graphical calculi, including string diagrams, Petri nets and reaction networks; - languages and frameworks, including process algebras, proof nets, type theory and game semantics; - abstract algebra and pure category theory, including monoidal category theory, higher category theory, operads, polygraphs, and relationships to homotopy theory; - quantum algebra, including quantum computation and representation theory; - tools and techniques, including rewriting, formal proofs and proof assistants, and game theory; - industrial applications, including case studies and real-world problem descriptions. -------------------------------------------------------------- IMPORTANT DATES All deadlines are 23:59 Anywhere on Earth. Submission deadline: Monday 25 October 2021 Author notification: Wednesday 3 November 2021 Symposium dates: Monday 13 and Tuesday 14 December 2021 -------------------------------------------------------------- SUBMISSION INSTRUCTIONS Submission are by EasyChair, via the SYCO 8 submission page: https://easychair.org/conferences/?conf=syco8 Submission is easy, with no format requirements or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere. Think creatively: you could submit a recent paper, or notes on work in progress, or even a recent Masters or PhD thesis. In the event that more good-quality submissions are received than can be accommodated in the timetable, the programme committee may choose to *defer* some submissions to a future meeting, rather than reject them. Deferred submissions can be re-submitted to any future SYCO meeting, where they will not need peer review, and where they will be prioritised for inclusion in the programme. Meetings will be held sufficiently frequently to avoid a backlog of deferred papers. If you have a submission which was deferred from a previous SYCO meeting, it will not automatically be considered for SYCO 8; you still need to submit it again through EasyChair. When submitting, append the words "DEFERRED FROM SYCO X" to the title of your paper, replacing "X" with the appropriate meeting number. There is no need to attach any documents. -------------------------------------------------------------- PROGRAMME COMMITTEE TBA Chair: Maaike Zwart, IT University of Copenhagen Local organisers: Amar Hadzihasanovic, Tallinn University of Technology Pawel Sobocinski, Tallinn University of Technology -------------------------------------------------------------- STEERING COMMITTEE Ross Duncan, University of Strathclyde Chris Heunen, University of Edinburgh Dominic Horsman, University of Oxford Aleks Kissinger, University of Oxford Samuel Mimram, Ecole Polytechnique Simona Paoli, University of Aberdeen Mehrnoosh Sadrzadeh, University College London Pawel Sobocinski, Tallinn University of Technology Jamie Vicary, University of Cambridge [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Casa abierta sobre Teoría de Categorías, 18-19/11/2021.
by Matias M 27 Sep '21

27 Sep '21

CAUTION: The Sender of this email is not from within Dalhousie. Dear all, We are happy to announce the Open House on Category Theory 2021. A short, informal meeting which ideally would have taken place in México but, because of the covid pandemic, will take place online; 18-19 November 2021. We will seize the occasion to celebrate Francisco 'Quico' Marmolejo's 60th birthday, so we particularly encourage talks on Pseudo-monads, distributive laws, Toposes and/or Axiomatic Cohesion. If you would like to assist, please send a message to matias.menni(a)gmail.com including your name and affiliation. If you would also like to give a talk, you need only include the title and abstract in your message. Further information may be found on the meeting's website. https://sites.google.com/view/catcat-2021-en/inicio Best regards, Matías Menni and Luis Turcio. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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***SPAM?*** categories: Re: Parts and telling apart
by Posina Venkata Rayudu 25 Sep '21

25 Sep '21

CAUTION: The Sender of this email is not from within Dalhousie. I just thought of adding that the conditions for the representability of a category C as a functor category B^A is given my Roos' theorem (kindly translated by Professor Lafforgue): Roos' Theorem, as stated on page 415 of SGA4 (volume 1), says that the three following conditions on a topos E are equivalent : i) The family of essential points of E is conservative. (NB : A point is called "essential" when its fiber functor not only has a right adjoint but also a left adjoint.) ii) The full sub-category of E consisting in objects which are connected - non empty and projective is generating. iii) E is a presheaf topos. At the end of the volume, there is a reference to three notes (in French) of Roos at "Comptes Rendus de l'Académie des Sciences", with the general title "Distributivity of colimits with respect to limits in toposes". This reference is to CR 259 (August and September 1964) : p. 969-972, 1605-1608 and 1801-1804. In the context of the mutual relationship between change/variation and unity/cohesion, I'd like to add: A monoid with a single constant (as a kind of variation) gives rise to a topos (of right actions of the monoid), i.e. idempotents (Ex. 5 in Conceptual Mathematics, p. 367; which is a quality type; Definition 1 in http://tac.mta.ca/tac/volumes/19/3/19-03.pdf) whose subobject classifier is not connected, and hence fails to satisfy one of the axioms of cohesion (Axiom 2 in http://tac.mta.ca/tac/reprints/articles/9/tr9.pdf) On the other hand, a monoid with two constants (another kind of variation) gives rise to a cohesive topos (reflexive graphs; Conceptual Mathematics, p. 367). With regard to UNIVERSAL MAPPING PROPERTY, it was also not discovered in its "simplest" (structureless) instantiation of initial set (as that which has exactly one function to any set); it required a more structured setting: Samuel, P. (1948) On universal mappings and free topological groups, Bull. A.M.S. 54: 591-598 (https://www.ams.org/journals/bull/1948-54-06/S0002-9904-1948-09052-8/S0002-…) Is it related to the difficulty of seeing function as a structure-preserving map, given that the structure preserved is structurelessness? Lastly, is the coseparator ~ subobject classifier (isomorphism, after discounting that the 'pointed' in pointed object definition of subobject classifier) valid in the discrete/constant subcategory of any topos (category with subobject classifier)? I eagerly look forward to your corrections. Thank you, posina On Fri, Sep 24, 2021 at 3:54 AM Posina Venkata Rayudu <posinavrayudu(a)gmail.com> wrote: > > Dear All, > > I hope and pray you and your families are all well. > > If I may, as I was preparing to give a lecture on SUBOBJECT > CLASSIFIER, as part of the Conceptual Mathematics seminar series at > Poornaprajna Institute of Scientific Research (https://ppisr.res.in/) > I started recollecting how much I loved the formula 2^|X| for the > number of subsets of a set X, when I first learned it in high school > (it was simple :-). I remember listing out all the eight subsets in > the case of X = {a, b, c}. (Grothendieck's profound insight of > defining subsets as 1-1 functions was definitely not part of my > understanding of SUBSET.) And about learning that the number of > functions from a domain set X to a codomain set Y is |Y|^|X|, and with > |Y| = 2 we could clearly see that: > > the number of subsets of a set X = the number of functions from the > set X to a two-element set 2 = 2^|X| > > It's not out of the realm of possibilities that I might have listed > all the functions from X to 2, which, along with all the subsets of X, > would have brought the 1-1 correspondence: > > parts of a set X > ---------------------------- > functions from X to 2 > > into a clear view. > > Be that as it may, what occluded SUBOBJECT CLASSIFIER from set > theorists and mathematical logicians; even Grothendieck missed it, but > in his characteristic kindness called it Lawvere element, upon > Professor F. William Lawvere's definition of subobject classifier as > part of his axiomatization of topos (please see p. 7, > http://www.mat.uc.pt/~picado/lawvere/interview.pdf) > > This is not an isolated incident in science; it appears to be a > pattern--wherein far-reaching constructions are not initially > conceptualized/recognized in their simplest instantiation, which is > where the figural salience of the concepts is clearly visible for all > to see and use--in scientific practice. A similar case can be made > about universal mapping properties (e.g. terminal set 1 = {*}) and > about category theory itself. That sets and functions form a > mathematical category is not easy to ignore, but category theory took > birth in an inaccessible realm rather remote for sets and functions > (thanks to the then prevalent practice of identifying a function with > its graph ((a, f(a)); cf. Conceptual Mathematics, pp. 293-294). > > Be that as it may, subset [and its representability by maps to > subobject classifier] is also related to the telling apart the figures > constituting the inside of an object of a category (with subobject > classifier or topos; see Sets for Mathematics, pp. 18-21). In the > case of sets, with coseparator 2 = {*, *} as the property type, there > are enough properties to tell apart any two different 1-shaped figures > (points/elements) in any set. Does subobject classifier (in every > category with it/topos) always serve as the property type coadequate > to tell apart figures in any object of the category/topos? What > difference does the fact that subobject classifier is defined as a > pointed object (map from the terminal object of the category to the > subobject classifier, i.e., t: 1 --> 2, where t(*) = t; see Exercise 8 > in Conceptual Mathematics, p. 337), which is not the case with > coseparator, i.e. it is just a constant/discrete/abstract 2-element > set. It seems a little odd that representing parts of an object > amounts to telling apart all different figures constituting the > object, simply going by the fact that PART (monomorphism) is a special > type of FIGURE (a morphism A --> B is an A-shaped figure in B; > Conceptual Mathematics, pp. 81-85; for additional context, among other > mathematical clarifications, see > https://cgasa.sbu.ac.ir/article_12425_b4ce2ab0ae3a843f00ff011b054f918b.pdf) > Telling apart figures in a domain object also appears to be the job of > epimorphisms (with subobject classifier of the category as codomain > object). > > Is > > coseparator ~ subobject classifier > > (isomorphism; after discounting that the 'pointed' in pointed object > definition of subobject classifier) specific to localic (as opposed to > cohesive) toposes (cf. Sets for Mathematics, pp. 93-94)? > > Also from a pedagogical perspective, given that our everyday > experience is that of categories of objects, all of which partake in > the essence/theory T characterizing a category C (Sets for > Mathematics, pp. 154-155), and since all objects of any category > partake in the essence T of the category, the transformations between > objects of the category necessarily preserve the essence, and hence > are structure-respecting maps (see Conceptual Mathematics, pp. > 149-151), which, in turn, are representable as natural transformations > (ibid, p. 378). Equally accessible is the idea that mathematical > objects, which are about everyday objects, are not unlike everyday > objects in that they are also made up of figures of various basic > shapes and their incidences (my attempt to introduce basic shapes and > their incidence relations to designers didn't get far; > https://zenodo.org/record/3924760#.YUzpptJBzZ4) And, then, in > addition to cohesion/essence/theory characterizing a category (in a > sense, we are limiting ourselves to presheaves or those categories C > that can be represented as contravariant functors M: T^op --> S, i.e. > as diagrams in the category S of sets), we can see that the natural > transformation (not unlike the transformations that we encounter in > our everyday life such as a water flowing downhill) respects the > essence of the object of a category that is being transformed (which > is T in the natural transformation of M: T^op --> S to N: T^op --> S). > Equally importantly, we can also begin with a mode of variation/change > and arrive at a category of objects (topos of right actions of a > monoid [objectifying change/variation]; Conceptual Mathematics, pp. > 360-361). This direction, i.e. objectification of change is > particularly important, given that we are given change (the basic > building block of our conscious experiences is contrast) and those > changes (natural transformations), in respecting/preserving the way > figures of various basic shapes stick together, make it possible to > reconstruct objects (as functors) based on the given change/variation > (but for natural transformations preserving the unity of objects > transformed, by the time I reach Malabar Cafe for ginger tea, there is > nothing stopping my leg being up in the clear skies of Bengaluru ;-) > > Is it inappropriate to claim that a universe of everyday experience is > accounted for by the notion of: > > categories of objects > > along with the mutual determination of: > > change/variation <--> cohesion/unity > > and does so in a manner accessible to total beginners (here's an > everyday objectification of change; https://youtu.be/r0kLC-pridI) > > Your time permitting, please correct any mistakes I might have made in > my characterization of categories of objects. > > I eagerly look forward to your corrections. > > Thanking you, > posina [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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ETAPS 2022 2nd joint call for papers
by Tarmo Uustalu 25 Sep '21

25 Sep '21

CAUTION: The Sender of this email is not from within Dalhousie. Why choose ETAPS? - ETAPS is one of the world's leading fora for research on software science, with a history of more than 25 years. - The proceedings of ETAPS appear in gold open access, with no article processing charge for the authors specifically. - ETAPS has low participation fees for all and for students in particular. New in 2022: - Like FASE for several years already, ESOP will also use double-blind review this time. - FASE has a new paper category of new ideas and emerging results (NIER) papers. - The pre-paper-acceptance artifact submission deadline for TACAS is now a little later than the paper submission deadline. - ESOP and FASE welcome voluntary submission of artifacts for evaluation after paper acceptance; the outcome will not change the paper acceptance decision. - Among the satellite events, there is a PhD student mentoring workshop. ****************************************************************** JOINT CALL FOR PAPERS 25th European Joint Conferences on Theory and Practice of Software ETAPS 2022 Munich, Germany, 2-7 April 2022 https://etaps.org/2022 ****************************************************************** -- ABOUT ETAPS -- ETAPS is the primary European forum for academic and industrial researchers working on topics relating to software science. ETAPS, established in 1998, is a confederation of four annual conferences, accompanied by satellite workshops. ETAPS 2022 is the twenty-fifth event in the series. -- MAIN CONFERENCES (4-7 April) -- * ESOP: European Symposium on Programming (PC chair: Ilya Sergey, Yale-NUS College and National University of Singapore, Singapore) * FASE: Fundamental Approaches to Software Engineering (PC chairs:Einar Broch Johnsen, University of Oslo, Norway, and Manuel Wimmer, Johannes Kepler Universität Linz, Austria) * FoSSaCS: Foundations of Software Science and Computation Structures (PC chairs: Patricia Bouyer, CNRS, LMF, France, and Lutz Schröder, FAU Erlangen-Nürnberg, Germany) * TACAS: Tools and Algorithms for the Construction and Analysis of Systems (PC chairs: Dana Fisman, Ben-Gurion University of the Negev, Israel, and Grigore Rosu, University of Illinois at Urbana-Champaign, USA) TACAS '22 will host the 11th Competition on Software Verification (SV-COMP). -- INVITED SPEAKERS -- * Unifying speakers: Alexandra Silva (University College London, UK / Cornell University, USA) Tomáš Vojnar (Brno University of Technology, Czechia) * FoSSaCS invited speaker: Nathalie Bertrand (Inria Rennes, France) * TACAS invited speaker: Lenore Zuck (University of Illinois at Chicago, USA) * Tutorial speakers: Stacey Jeffery (CWI and QuSoft, The Netherlands) a further tutorial speaker tba -- IMPORTANT DATES * Paper submission (also, pre-paper-acceptance artifact registration (TACAS)): 14 October 2021 23:59 AoE * Pre-paper-acceptance artifact submission (TACAS): 4 November 2021 23:59 AoE * Rebuttal (ESOP, FoSSaCS and, partially, TACAS): 7 December 00:01 AoE - 9 December 2021 23:59 AoE * Paper notification: 23 December 2021 * Post-paper-acceptance artifact submission (ESOP, FASE, TACAS): 5 January 2022 23:59 AoE * Paper final versions: 26 January 2022 * Artifact notification: 16 February 2022 -- SUBMISSION INSTRUCTIONS -- The four main conferences of ETAPS 2022 solicit contributions of the following types. All page limits are given **excluding the bibliography**. * ESOP: regular research papers of max 25 pp * FASE: regular research papers and empirical evaluation papers of max 18 pp, new ideas and emerging results (NIER) papers of max 8 pp, tool demonstration papers of max 6 pp (+ mandatory appendix of max 6 pp), * FoSSaCS: regular research papers of max 18 pp * TACAS: regular research papers, case study papers and regular tool papers of max 16 pp, tool demonstration papers of max 6 pp For definitions of the different paper types and specific instructions, where they are present, see the webpages of the individual conferences. All accepted papers will appear in the proceedings and have presentations during the conference. A condition of submission is that, if the submission is accepted, one of the authors attends the conference to give the presentation. We plan ETAPS 2022 as a hybrid conference; remote attendance and presentation will be possible. Submitted papers must be in English presenting original research. They must be unpublished and not submitted for publication elsewhere. In particular, simultaneous submission of the same contribution to multiple ETAPS conferences is also forbidden. Submissions must follow the formatting guidelines of Springer's LNCS (use the llncs.cls class) and be submitted electronically in pdf through the Easychair author interface of the respective conference. Submissions not adhering to the specified format and length may be rejected immediately. ESOP and FASE will use **double-blind reviewing**. Authors are asked to omit their names and institutions; refer to prior work in the third person, just as prior work by others; not to include acknowledgements that might identify them. ESOP and FoSSaCS will use an **author rebuttal phase**. TACAS will use rebuttal for selected submissions (those in the gray zone). Artifact submission and evaluation Regular tool paper and tool demonstration paper submissions to TACAS must be accompanied by an artifact submitted shortly after the paper. The artifact will be evaluated and the outcome will be taken into account in the acceptance decision of the paper. For research paper and case study paper submissions, pre-paper-acceptance submission of an artifact is optional; if an artifact is submitted at this point, it will be handled like described above. Alternatively, authors of papers of these categories may submit an artifact for evaluation after the paper has been accepted. The outcome of the artifact evaluation will then not change the paper acceptance decision. ESOP and FASE will also have artifact evaluation, but participation in it is voluntary; the artifact submission deadline is after the paper notification deadline. The outcome will not alter the paper acceptance decision. For specific instructions regarding artifacts, see the webpages of the individual conferences. -- PUBLICATION The proceedings will be published in the Advanced Research in Computing and Software Science (ARCoSS) subline of Springer's LNCS series. The proceedings volumes will appear in gold open access, so the published versions of all papers will be available for everyone to download from the publisher's website freely, from the date of online publication, perpetually. The copyright of the papers will remain with the authors. -- BEST PAPER AWARDS The strongest papers of the four conferences will be nominated for the ETAPS best paper awards of EAPLS, EASST and EATCS. The ETAPS test of time award recognizes outstanding papers published at ETAPS more than 10 years in the past. -- SATELLITE EVENTS (2-3 April) -- A number of satellite workshops and other events will take place before the main conferences: CMCS, COOP, HCVS, LiVe, MARS, MSFP, PLACES, QAVS, RRRR, Rust Workshop, SynCop, VerifyThis, VPT, WRLA Also in the satellite events program is a Mentoring Workshop. For closer information, check https://etaps.org/2022/workshops . -- CITY AND HOST INSTITUTION -- Munich, Germany, is the capital and most populous city of Bavaria. Nowadays, the city is a global center of art, science, technology, finance, publishing, culture, innovation, education, business, and tourism. It is home to Ludwig Maximilian's University (LMU) and Technische Universität München (TUM), many scientific institutions, and world-class technology and science museums such as Deutsches Museum and BMW Museum. Iconic places to visit in Munich include the Munich Residenz, Marienplatz, the Old Town Hall, the famous Glockenspiel on the New Town Hall, the Frauenkirche, the English Garden, the Olympic Park and Nymphenburg Palace. The conference will be hosted by Technische Universität München. -- ORGANIZERS -- General chair: Jan Křetínský (Technische Universität München, Germany) Workshops chair: Dirk Beyer (Ludwig-Maximilians Universität München, Germany) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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***SPAM?*** categories: Reminder: 2 year position on HoTT and related topics in Stockholm, deadline Friday next week
by Anders Mortberg 24 Sep '21

24 Sep '21

CAUTION: The Sender of this email is not from within Dalhousie. Dear all, A quick reminder of the postdoctoral position in HoTT and related topics that I advertised a couple of weeks ago, with deadline *October 1* (midnight Stockholm time). Full details and application at: https://www.su.se/english/about-the-university/work-at-su/available-jobs?rm… Departmental webpage: https://www.math.su.se/english/research Once again, please get in touch with me if you have any questions about the position or application process! Best, Anders On Mon, Sep 6, 2021 at 1:43 PM Anders Mortberg <anders.mortberg(a)math.su.se> wrote: > Dear all, > > I’m pleased to announce that we’re hiring a postdoctoral researcher in > homotopy type theory and related topics at Stockholm University. It’s a 2 > year position, provisionally starting January 2022, in the computational > mathematics group of the Mathematics Department. The application deadline > is 1 October. > > We welcome all applicants interested in working on homotopy type theory > and related topics. Potential project topics include, but are not limited > to: homotopy type theory, categorical models of type theories, cubical type > theories, implementation of proof assistants, computer formalization of > mathematics and computer science. > > Full details and application at: > https://www.su.se/english/about-the-university/work-at-su/available-jobs?rm… > > Departmental webpage: https://www.math.su.se/english/research > > Please get in touch with me if you have any questions about the position! > > Best, > Anders > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Oct. Scientific American
by Michael Barr, Prof. 24 Sep '21

24 Sep '21

CAUTION: The Sender of this email is not from within Dalhousie. It contains a feature article on category theory by none other than Emily Riehl! Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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