Re: Reference requested
Thanks everybody for comments, although I guess the use goes so far back into antiquity that the request for an original reference is unanswerable. For context, with two young collaborators (Bertrand Guillou and Mona Merling), I have a draft in progress tentatively entitled ``Chaotic categories and equivariant classifying spaces''. I prefer `chaotic' to `indiscrete' not just because of the `coarse' implications of the latter, but because indiscrete spaces are boring, `null or banal', whereas chaotic categories have genuinely significant applications. They are quite surprisingly central to the theory of universal bundles, equivariant or not. Via the (product-preserving) classifying space construction from categories (especially categories internal to spaces) to spaces, they provide a rich source of contractible spaces that can very easily be given interesting additional structure. That is just what one wants when constructing universal bundles. More fun, it is just what one wants to construct an E infinity operad of G-categories that defines `genuine' symmetric monoidal G-categories (which are not merely symmetric monoidal categories on which a group G acts in the obvious `naive' way). These which give rise to `genuine' G-spectra. Genuine G-spectra that define equivariant algebraic K-theory arise in precisely this way. All starting from chaotic trivialities. Cheers, Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
To may@math.uchicago.edu, categories Sender: categories@mta.ca Precedence: bulk Reply-To: F William Lawvere <wlawvere@buffalo.edu> Probably it was not coined but borrowed, from general topology. For at least 50 years, the two, words chaotic codiscrete were alternate terminology for a certain kind of space. I prefer "codiscrete" since it clearly indicates something opposite to discrete, the precise sense of oppositeness being that of inclusions adjoint to the same uniting functor, often called the "underlying". (In higher dimensions, coskeletal and skeletal are similar identical opposites, with "truncation" as uniter). In fact every groupoid is a colimit of codiscrete ones, indeed groupoids form a reflective subcategory of the topos that classifies Boolean algebras, and the latter has a site consisting of codiscrete groupoids. (The generic Boolean algebra 2^( ) has as its natural geometric realization the infinite-dimensional sphere, containing the ordinary interval as a generating distributive lattice). More recently, "chaotic" has come to have a different meaning, although one also involving a right adjoint. If f:X->Y is a map from a space equipped with an action of a monoid T to another space, then f is a chaotic observable if the induced equivariant map from X to the cofree action Y^T is epimorphic. A classic "symbolic" example has Y=pi0(X), i.e. the observation recorded by f is merely of which component we are passing through, but almost any T-sequence of such is obtained by a sufficiently clever choice of initial state in X. Bill Lawvere
Date: Wed, 28 Sep 2011 20:35:02 -0500 From: may@math.uchicago.edu CC: categories@mta.ca Subject: categories: Reference requested
I have a reference question. Who first coined the term ``chaotic category'' for a groupoid with a unique morphism between each pair of object, and in what context? It is a ridiculously elementary concept, but one that is extremely useful in work on equivariant bundle theory that is needed for equivariant infinite loop space theory and equivariant algebraic K-theory.
Peter May
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Bill recalls:
For at least 50 years, the two, words chaotic codiscrete were alternate terminology for a certain kind of space.
My memory rather matches instead what I see in (3.2(d)) of Willard, that the topology with only the whole space and the empty set "open" is called either 'trivial' or 'indiscrete'. In my experience I've never encountered 'chaotic' as the adjective used for that attribute -- indeed, 'chaotic' would have conflicted rather badly with the Chaos Theory arising out of René Thom's Catastrophe Theory of the early '60s or so -- and 'codiscrete' strikes me as what only a categorist hoping (as we many of us long did) to systematize terminology into dual camps of 'properties' and 'coproperties' (in the model of adjoint/coadjoint, limit/colimit, terminal/coterminal, etc.) could have come up with -- not a term any self-respecting point-set topologist would have thought to use :-) . 'Codiscrete' of course does, for just that reason, have its merits, as Bill points out:
I prefer "codiscrete" since it clearly indicates something opposite to discrete, the precise sense of oppositeness being that of inclusions adjoint to the same uniting functor, often called the "underlying". (In higher dimensions, coskeletal and skeletal are similar identical opposites, with "truncation" as uniter).
In fact every groupoid is a colimit of codiscrete ones, indeed groupoids form a reflective subcategory of the topos that classifies Boolean algebras, and the latter has a site consisting of codiscrete groupoids. (The generic Boolean algebra 2^( ) has as its natural geometric realization the infinite-dimensional sphere, containing the ordinary interval as a generating distributive lattice).
And I object not one whit to any of that :-) .
More recently, "chaotic" has come to have a different meaning, although one also involving a right adjoint. If f:X->Y is a map from a space equipped with an action of a monoid T to another space, then f is a chaotic observable if the induced equivariant map from X to the cofree action Y^T is epimorphic. A classic "symbolic" example has Y=pi0(X), i.e. the observation recorded by f is merely of which component we are passing through, but almost any T-sequence of such is obtained by a sufficiently clever choice of initial state in X.
This again suggests that 'chaotic' might not be the best choice of adjective for that indiscrete/codiscrete topology, or the analogous type of category, or groupoid, or topological category or groupoid. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Fred, you caught me ! Indeed my memory of JL Kelley was blurred by wishful thinking. Hence, dear friends, please be so generous as to ignore the attempt at historical justification and instead further elaborate the rational arguments as Fred has. Bill Lawvere On Fri 09/30/11 5:19 PM , "Fred E.J. Linton" fejlinton@usa.net sent:
Bill recalls:
For at least 50 years, the two, words chaotic codiscrete were alternate terminology for a certain kind of space. My memory rather matches instead what I see in (3.2(d)) of Willard, that the topology with only the whole space and the empty set "open"is called either 'trivial' or 'indiscrete'.
In my experience I've never encountered 'chaotic' as the adjective used for that attribute -- indeed, 'chaotic' would have conflicted rather badly with the Chaos Theory arising out of René Thom's Catastrophe Theory of the early '60s or so -- and 'codiscrete' strikes me as what only a categorist hoping (as we many of us long did) to systematize terminology into dual camps of 'properties' and 'coproperties' (in the model of adjoint/coadjoint, limit/colimit, terminal/coterminal, etc.) could have come up with -- not a term any self-respecting point-set topologist would have thought to use :-) .
'Codiscrete' of course does, for just that reason, have its merits, as Bill points out:
I prefer "codiscrete" since it clearly indicates something> opposite to discrete, the precise sense of oppositeness being> that of inclusions adjoint to the same uniting functor, often called> the "underlying". (In higher dimensions, coskeletal and skeletal> are similar identical opposites, with "truncation" as uniter).> In fact every groupoid is a colimit of codiscrete ones, indeed> groupoids form a reflective subcategory of the topos that classifies > Boolean algebras, and the latter has a site consisting of codiscrete> groupoids. (The generic Boolean algebra 2^( ) has as its natural> geometric realization the infinite-dimensional sphere, containing > the ordinary interval as a generating distributive lattice). And I object not one whit to any of that :-) .
More recently, "chaotic" has come to have a different meaning, > although one also involving a right adjoint. If f:X->Y is a map > from a space equipped with an action of a monoid T to another> space, then f is a chaotic observable if the induced equivariant> map from X to the cofree action Y^T is epimorphic. A classic "symbolic"> example has Y=pi0(X), i.e. the observation recorded by f is merely of> which component we are passing through, but almost any > T-sequence of such is obtained by a sufficiently clever choice> of initial state in X.
This again suggests that 'chaotic' might not be the best choice of adjective for that indiscrete/codiscrete topology, or the analogous type of category, or groupoid, or topological category or groupoid.
Cheers, -- Fred
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Peter May, in re the Subject: categories: Re: Reference requested, wrote
... I prefer `chaotic' to `indiscrete' not just because of the `coarse' implications of the latter, but because indiscrete spaces are boring, `null or banal', whereas chaotic categories have genuinely significant applications. ...
Be that as it may, I sought Search-engine advice regarding the use of the 'chaotic topological space' lingo, and came up with the following 'hits', of which only the first reflects, in an afterthought, Peter's usage, while the others all envision something rather quite different: 1) From http://en.wikipedia.org/wiki/Grothendieck_topology : The discrete and indiscrete topologies Let C be any category. To define the discrete topology, we declare all sieves to be covering sieves. If C has all fibered products, this is equivalent to declaring all families to be covering families. To define the indiscrete topology, we declare only the sieves of the form Hom(−, X) to be covering sieves. The indiscrete topology is also known as the biggest or chaotic topology, and it is generated by the pretopology which has only isomorphisms for covering families. A sheaf on the indiscrete site is the same thing as a presheaf. Other uses of 'chaotic', having nothing to do with indiscreteness, predominate: 2) From http://www.math.uh.edu/~hjm/pdf26%284%29/03chara.pdf , reflecting the content of ON GENERALIZED RIGIDITY by JANUSZ J. CHARATONIK from Houston Journal of Mathematics (© 2000 University of Houston) Volume 26, No. 4, 2000 : A nondegenerate topological space X is said to be: (a) chaotic if for any two distinct points p and q of X there exists an open neighbourhood U of p and an open neighbourhood V of q such that no open subset of U is homeomorphic to any open subset of V ; ... [snip] ... 3) CHAOTIC GROUP ACTIONS www.math.zju.edu.cn/amjcu/B/200301/030108.pdf ... no chaotic group actions on any topological space with free arc. ... ... topological space which admits a chaotic group action but admits ... 4) CHAOTIC POLYNOMIALS IN SPACES OF CONTINUOUS AND ... personales.upv.es/almimon/Preprint%20Aron-Miralles.pdf ... show that there exist chaotic homogeneous polynomials of degree m ≥ 2. ... So I'd imagine 'chaotic', for 'indiscrete', is best dropped, and either 'indiscrete', 'codiscrete', or 'trivial' be used instead. NB: while it's true that the trivial (indiscrete) topology on a set X is initial, in the sense that, as a collection of subsets of X, it's the smallest that's a topology on X, the indiscrete topological space on X is terminal among all topological spaces on X and mappings that restrict to the identity on X; the trivial (indiscrete) pre-order on X is likewise terminal, in the sense that, as a subset of X x X, it's the largest. A connected pre-ordered groupoid (i.e., indiscrete category), being equivalent to the terminal category 1, has the property that, for each category X, it admits exactly one isomorphism class of functor from X, but while that may make it 2-terminal or [( co | op ) lax-] terminal, I'd still probably prefer to avoid such ... umm ... terminalogy :-) . Cheers, -- Fred PS: I re-emphasize: of all the hits I found, only one amongst the first two dozen -- the first cited above -- spoke of the trivial topology as the chaotic topology; ALL the others used 'chaotic' in some other way, DESPITE the search having been explicitly for [ chaotic topological space ]. And there were "about 175,000 results" all told :-) . -- F. PPS: Typos? Perhaps; please forgive, I couldn't spiel-chuck. -- F. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I was very happy and relieved when reading Fred Linton's message underneath and then Bill's agreement, since, after reading Ross Street's message about the use of the term "chaotic" by Bourbaki and French culture, I had started getting serious doubts about the state of my own memory. Indeed everybody can check that, from Bourbaki's first edition of Topology to the final one (seriously reshaped by Grothendieck, I think), published in 1971 and 74, though the indiscrete topology is the first example given for a topology, followed by the discrete one, it is (rather curiously) not given a name. However, at least from the fifties, and certainly before, the French terminology "topologie grossie`re" was universally accepted and used as the unique one by the whole community of French mathematicians, at all levels, from education to research. I am not aware that things have changed, though it is nowadays difficult to find a text written in French by a young French native speaker which would not be a poor translation from Frenglish of a poor translation from French to Frenglish (a consequence of the fact that French language is no longer taught at French schools, an open secret ; well, you understand that I am not a candidate to the Ministry of Education). Of course (!) the use of the English term "coarse", which is perhaps the translation in down-to-earth language of the French "grossier" was much less successful, and probably used essentially in texts which were more or less translated from French or deriving from French culture. It seems to be more or less abandonned nowadays and to have got some very special quite different and technical meaning (in connection with the study of Gromov metrics). It seems also that the English-speaking classical topologists are nowadays generally pleased with the neologism/pun pair offered by in(or un-)discrete/indiscreet, which cannot have any French equivalent, for orthographic reasons (just one possible orthograph and meaning for "indiscret"). As to the term "chaotic", I have exactly the same (absence of) experience as Fred (perhaps it was used by Grothendieck's school in the sixties, but I am not familiar with that, and don't remember to have heard of it) and think it is evoking unavoidably the field of dynamical systems (where however it is probably used without a very precise and unique definition). I'm sure I'll never use such a term for naming the very simple notion which is called "pair groupoid" in the book by Kirill on Lie groupoids ; this latter terminology (which however I don't like very much, see further) is presently accepted by many authors, and I'm surprised that none of them came to the fore in that discussion. It should be noted however that the geometer van Est sometimes used the term "blackhole" for naming the space of leaves of a foliation when its topology (in the elementary sense, i.e. the quotient one) is the indiscrete one; I suppose this was evocating the very chaotic behaviour of the leaves in such a case. However all the preceding (purely linguistic) considerations are very secondary in my opinion and I turn now in more details to my main point which was expressed in my previous message but found no echo. First of all I claim that I perfectly agree with Peter May about the basic importance of what I prefer to call banal groupoids (in some internal contexts I am currently using, they don't always exist, an a quest for suitable substitutes may be an important motivation). But I don't think this justifies to look for terrific names, which anyway can reflect but a tiny part of these important properties. The term "banal" should not be regarded as pejorative. Think to mathematical objects like 0, 1, 2: they are probably among the most banal, or even silly, objects from a naïve point of view, though they are bearing a lot of very rich but highly degenerate structures, and they are perhaps encapsulating some basic processes for explaining the transition from nothingless to being (!); this would not be a reason for trying to give them mysterious names borrowed from philosophy or theology (!). More precisely the adjunction property which is intended to be stressed by introducing duets such as discrete/co-discrete, is just a part of an iceberg of some interesting adjunction strings (with different lengths) arising from various forgetful functors. But the structure of these strings are strongly depending on the "structure" you are forgetting and are quite different when dealing with the algebraic structure of groupoid or with a topological structure; in the latter case you may get the pi_0 functor (connected components) as arising in that way in such a string. Now when dealing with topological (and particularly smooth) groupoids (internal groupoids in the category Top or Dif), subtle interesting interactions and clashes arise between the two strings: in a certain sense it may be said than a suitable pi_1 functor emerges as a certain kind of internal pi_0, in a way that includes notably the correct pi_1 concept (Poincare' functor) for spaces of leaves of a foliation; the latter was introduced by different geometers by means of various ad hoc constructions (Haefliger, van Est, Paul ver Ecke); it turns out that these constructions may be derived from a general algebraic construction described in the book by Gabriel-Zisman, which introduces an adjoint for the "nerve functor" from groupoids to simplicial sets. This approach yields immediately a van Kampen theorem by preservation of colimits, according to Ronnie's presentation. This whole stuff was described in a CRAS Note (Paris) which I published in 1989 (cosigned with my irakian student Alta'ai) (t. 309, Se'rie I, p. 503-506). Unhappily neither Haefliger nor van Est understood a single word of this Note (which indeed they tried to get rejected, unsuccessfully), since they are not at all categorically minded. The main result was formulated in a different way, using what I called simplified calculus of fractions for Morita equivalences (which cannot be derived as a special case from the classical Gabriel-Zisman calculus of fractions) in a paper published in les Cahiers de Topologie in the same year, nowadays available with minor corrections and important added comments in arXiv:0803.4209v1. In fact my secret hope is that such a construction (from pi_0 to pi_1) might be a first step for an inductive process leading to higher homotopy in the spirit of Ronnie and Higgins. Clearly the use of the same word "discrete", originated from (elementary) Topology to describe adjointness properties which are quite different for groupoids and for topological spaces, but are able to lead to fundamental interactions when studied for topological groupoids, make it strictly impossible to perform that type of studies, which I regard as very fecund. More generally further comments about the widespread misuse of topological terminology in the purely algebraic framework of abstract groupoids may be found in arXiv:0711.1608v1, a paper written on the occasion of Ehresmann's birthday 100th anniversary. ----- Message d'origine ----- De : "Fred E.J. Linton" <fejlinton@usa.net> À : <categories@mta.ca> Cc : "F. William Lawvere" <wlawvere@buffalo.edu>; "Peter May" <may@math.uchicago.edu> Envoyé : vendredi 30 septembre 2011 23:19 Objet : categories: Re: Reference requested Bill recalls:
For at least 50 years, the two, words chaotic codiscrete were alternate terminology for a certain kind of space.
My memory rather matches instead what I see in (3.2(d)) of Willard, that the topology with only the whole space and the empty set "open" is called either 'trivial' or 'indiscrete'. In my experience I've never encountered 'chaotic' as the adjective used for that attribute -- indeed, 'chaotic' would have conflicted rather badly with the Chaos Theory arising out of René Thom's Catastrophe Theory of the early '60s or so -- and 'codiscrete' strikes me as what only a categorist hoping (as we many of us long did) to systematize terminology into dual camps of 'properties' and 'coproperties' (in the model of adjoint/coadjoint, limit/colimit, terminal/coterminal, etc.) could have come up with -- not a term any self-respecting point-set topologist would have thought to use :-) . 'Codiscrete' of course does, for just that reason, have its merits, as Bill points out:
I prefer "codiscrete" since it clearly indicates something opposite to discrete, the precise sense of oppositeness being that of inclusions adjoint to the same uniting functor, often called the "underlying". (In higher dimensions, coskeletal and skeletal are similar identical opposites, with "truncation" as uniter).
In fact every groupoid is a colimit of codiscrete ones, indeed groupoids form a reflective subcategory of the topos that classifies Boolean algebras, and the latter has a site consisting of codiscrete groupoids. (The generic Boolean algebra 2^( ) has as its natural geometric realization the infinite-dimensional sphere, containing the ordinary interval as a generating distributive lattice).
And I object not one whit to any of that :-) .
More recently, "chaotic" has come to have a different meaning, although one also involving a right adjoint. If f:X->Y is a map from a space equipped with an action of a monoid T to another space, then f is a chaotic observable if the induced equivariant map from X to the cofree action Y^T is epimorphic. A classic "symbolic" example has Y=pi0(X), i.e. the observation recorded by f is merely of which component we are passing through, but almost any T-sequence of such is obtained by a sufficiently clever choice of initial state in X.
This again suggests that 'chaotic' might not be the best choice of adjective for that indiscrete/codiscrete topology, or the analogous type of category, or groupoid, or topological category or groupoid. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I would like to take up on one small point in Jean Pradines' excellent email on terminology. He also writes On 02/10/2011 15:48, jpradines wrote:
Gabriel-Zisman, which introduces an adjoint for the "nerve functor" from groupoids to simplicial sets. This approach yields immediately a van Kampen theorem by preservation of colimits, according to Ronnie's presentation.
I agree with the words `a van Kampen theorem', with emphasis on `a'. What seems to be little appreciated is the notion of fundamental groupoid on a set A of base points; for a simplicial set K this notion is not seen as the set of base points is usually thought of as K_0. The whole point of the more general construction is to choose A according to the geometry at hand (see a Grothendieck quote on my web page for `Topology and Groupoids'); for the topological circle one conveniently chooses A to consist of 2 points. The general situation with covers requires A to meet every path component of every 1,2,3 fold intersection of the sets of the cover, so this format of the theorem does not follow immediately from adjoint functor arguments. The higher dimensional theorems of van Kampen type have NOT been shown to follow from simple adjoint functor arguments, or indeed any simple arguments, but have led to extensive (but restricted) non abelian colimit calculations in higher homotopy. The discussion of codiscrete or banal (or .....) groupoids raises the question of who was first to see the banal groupoid on {0,1} as a kind of unit interval groupoid? This use was made in my 1968 book `Elements of modern topology', now reissued as T&G. I should mention Dedecker, Paul; Valderrama, Jerko Graphes et cographes sur une catégorie abstraite. Application à l'homotopie. C. R. Acad. Sci. Paris Sér. A-B 262 1966 A377–A380. From the review: "The notion of cograph leads to a definition of homotopy in a category". Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I recently reported that
... I sought Search-engine advice regarding the use of the 'chaotic topological space' lingo, and came up with ...
... surprisingly little. Early, early this morning I tried that again, but enclosing the search string in double-quotation marks. Ah, now I found two other references, worth citing, perhaps: 1) Volker Runde's 2005 Springer Universitext, isbn=038725790X, A Taste of Topology, includes a passage (page 72), beginning "Let (X,TX) be a chaotic topological space (ie, TX = {∅,X}), let (Y,TY) be a Hausdorff space, and let f: X → Y be continuous" and deducing such f must be constant; links (to Google Books and a PDF): [long url omitted by moderator], ftp://210.45.114.81/math/2007_07_06/Universitext/V.Runde%20A%20Taste%20of%20Topology.pdf (no hint, though, how 'standard' Runde thought his use of "chaotic" here was :-{ ); and 2) Mat{´ı}as Menni's 2000 Edinburgh PhD thesis {Exact Completions and Toposes} makes multiple mention of chaotic structures, with frequent citations of Bill Lawvere's interest in such things. All of Chapter 7 is about "Chaotic Situations", with Section 7.1 focused in particular on "Chaotic Objects"; and Section 8.4 returns to "Chaotic Situations". A hint of the flavor is given in the Introduction already: "In 1999, Longley introduced a typed version of the notion of a partial combinatory algebra in [68] and described how to build a category of assemblies Ass(A) over a [sic] such a structure A. Shortly after, Lietz and Streicher showed that the ex/reg completion of Ass(A) is a topos if and only if the typed structure A is equivalent, in a suitable sense, to an untyped structure. Their proof uses the notion of a generic mono (a mono τ such that every other mono arises as a pullback of τ along a not necessarily unique map) and of the constant-objects embedding of Set into the category Ass(A) which they see as an inclusion of codiscrete objects. Related to this, it should be mentioned that Lawvere had already advocated for a conceptual use of codiscrete or chaotic objects in other areas of mathematics (see for example [59, 55, 61, 63])." No surprise, then, to see the right adjoint ∇ to the forgetful functor Set -> Top described (p. 23) as follows: " ... the functor ∇: Set -> Top assigns to each set S the “chaotic” topological space with underlying set S and, as open sets, only S itself and the empty set." Cf. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.22.9817&rep=rep1&type=pdf --- BTW, those four Lawvere references are these: [55] F. W. Lawvere. Toposes generated by codiscrete objects in combinatorial topology and functional analysis. Notes for colloquium lectures given at North Ryde, New South Wales, Australia on April 18, 1989 and at Madison USA, on December 1, 1989. [59] F. W. Lawvere. Categories of spaces may not be generalized spaces as exemplified by directed graphs. Revista colombiana de matem´aticas, 20:179– 186, 1986. [61] F. W. Lawvere. Some thoughts on the future of category theory. In Proceed- ings of Category Theory 1990, Como, Italy, volume 1488 of Lecture notes in mathematics, pages 1–13. Springer-Verlag, 1991. [63] F. W. Lawvere. Unit and identity of opposites in calculus and physics. Applied categorical structures, 4:167–174, 1996. --- All told, eight hits, all either these two, or references to them, or search-database errors :-) . Not very heavy evidence in favor of "chaotic". So: cheers -- and back to [co-|in-]discrete, I fear :-) , -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
F William Lawvere -
Fred E.J. Linton -
jpradines -
Peter May -
Ronnie Brown