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January 19, 2007

More on Duality

Posted by David Corfield

Continuing our earlier discussion about duality, it’s worth noting a distinction that Lawvere and Rosebrugh introduce in chapter 7 of their Sets for Mathematics between ‘formal’ and ‘concrete’ duality. Formal duality concerns mere arrow reversal in the relevant diagrams, so

of course if the original diagrams had been given specific interpretation in terms of specific sets and mappings, such interpretation is lost when we pass to this formal dual in that the formal dualization process in itself does not determine specific sets and specific mappings that interpret the dualized statement. (p. 121)

Concrete duality, on the other hand, occurs in situations where a new diagram is formed from an old one by exponentiating each object with respect to a given dualizing object, e.g., XX becomes V XV^X, with VV the dualizing object. The arrows are naturally reversed in the new diagram.

Now,

Not every statement will be taken into its formal dual by the process of dualizing with respect to VV, and indeed a large part of the study of mathematics

space vs. quantity

and of logic

theory vs. example

may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite VV correspond or fail to correspond. (p. 122)

Very relevant for concrete dualities is Peter Johnstone’s Stone Spaces, especially chapter 6 and its discussion of schizophrenic objects. For a hard-hitting review by Johnstone of some Universal algebraists’ attempt to treat duality while minimizing contact with category theory, see this.

Schizophrenic objects have found a role in clarifying ω\omega-categorical issues. See Makkai and Zawadowski’s Duality for Simple ω\omega-Categories and Disks.

Posted at January 19, 2007 9:10 AM UTC

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Re: More on Duality

In connection with these formal and concrete dualities, I can’t resist mentioning the Chu construction and especially Chu spaces: Vaughn Pratt has emphasized that many familiar concrete dualities are embodied in the category of Chu spaces.

In a nutshell, the Chu construction is a way of constructing a *-autonomous category from the data of a symmetric monoidal closed category M with pullbacks and an object D of M. An object of Chu(M, D) is a triple (A, B, p) where p is a D-valued pairing between A and B, and a morphism (A, B, p) –> (A’, B’, p’) is a pair of morphisms f: A –> A’, g: B’ –> B which are transposes of one another with respect to the pairings p and p’. If you write down a formal description of these hom-sets in terms of pullback diagrams involving hom-sets of M, then you can more or less guess what the closed structure of Chu(M, D) looks like, by internalizing these homs. The dual of (A, B, p) is obtained by switching A and B, and the dualizing object is the pair (D, I) with the obvious pairing.

The category of Chu spaces is Chu(Set, 2). A nice point emphasized by Pratt is that for many instances of concrete dualities where the underlying set of the schizophrenic object is 2, the category and its opposite embed as dual subcategories in the self-dual category of Chu spaces. This applies in particular to Stone duality, the duality between ordinals and intervals [emphasized by Joyal in connection with his category of disks], the self-duality of sup-lattices, and many others that you can think of.

Posted by: Todd Trimble on January 19, 2007 4:04 PM | Permalink | Reply to this

Re: More on Duality

Does one ever find duality between bicategories arising from an object having two ‘commuting’ structures? I mean is it ever the case that something like the category of sets can be seen as possessing two structures, and so be used schizophrenically?

Posted by: David Corfield on January 19, 2007 9:40 PM | Permalink | Reply to this

Re: More on Duality

Judging by our experience with objects with two commuting structures e.g. groups, what kind of categories woudl permit such?

Posted by: jim_stasheff on August 3, 2013 1:58 PM | Permalink | Reply to this

Re: More on Duality

That’s an old comment of mine! Of course, it turned out that SetSet can act as a dualizing object. See, e.g., Forssell and Awodey’s duality between first order theories and their models, reported here.

Posted by: David Corfield on August 3, 2013 2:14 PM | Permalink | Reply to this

Re: More on Duality

In some cases this involves certain limits commuting with certain colimits. For example, SetSet is finitely complete and also filtered-colimit cocomplete, and finite limits commute with filtered colimits. So SetSet plays a role of dualizing object where on the one side we consider finitely complete categories and finitely continuous functors, and on the other complete and filtered-cocomplete categories, and this leads ultimately to the Gabriel-Ulmer duality between small finitely complete categories and locally finitely presentable categories. As we know, there are many variations on this theme.

Posted by: Todd Trimble on August 3, 2013 4:16 PM | Permalink | Reply to this

Re: More on Duality

I wonder what variants will be found for \infty-Groupoid as a dualizing object.

Posted by: David Corfield on August 4, 2013 7:44 AM | Permalink | Reply to this

Terminology

I’ve said this before and I’ll say it again: we need to find a word other than “schizophrenic”.

Groups that work with those who have schizophrenia are constantly trying to persuade lazy journalists etc. not to use the word in a way that perpetuates the simplistic comedy stereotype of a “split personality”. The mathematical usage is in exactly this vein. We should choose a word that doesn’t reinforce a misleading cliche about a serious illness.

I put this point at a category theory meeting a few years ago and other people (including Peter Johnstone) agreed. There were various suggestions for alternatives, but I can’t remember them. Maybe “ambiguous”, “ambivalent”, or “two-faced”? Ideas, anyone?

Posted by: Tom Leinster on January 19, 2007 10:58 PM | Permalink | Reply to this

Re: Terminology

Maybe “ambiguous”, “ambivalent”, or “two-faced”?

Those are all nice, but my favourite is ‘ambivalent’.

Posted by: Toby Bartels on January 20, 2007 12:21 AM | Permalink | Reply to this

Re: Terminology

How about ‘Janusian objects’ after the Roman God Janus?

I see ‘Janusian’ has been used to describe a form of thinking:

Janusian thinking—“actively conceiving two or more opposite or antithetical ideas, concepts, or images simultaneously,” according to the author’s definition—is proposed as a specific thought process that operates in the act of creation.

People also say ‘Janus-like’.

Posted by: David Corfield on January 20, 2007 11:58 AM | Permalink | Reply to this

Re: Terminology

David wrote:

How about ‘Janusian objects’ after the Roman God Janus?

Oddly enough, I thought of that as I was writing my previous comment, but dismissed it as too fanciful. I think it’s a more accurate word than any of the others, though it’s a bit unusual.

Posted by: Tom Leinster on January 20, 2007 12:21 PM | Permalink | Reply to this

Re: Terminology

Mathematicians seem more conservative than physicists when it comes to naming. But Alan Weinstein’s explanation for his ‘hopfish algebras’, see Friday 21 October entry from my old blog shows some humour is possible.

Posted by: David Corfield on January 20, 2007 12:32 PM | Permalink | Reply to this

Re: Terminology

David wrote:

Mathematicians seem more conservative than physicists when it comes to naming.

Really? What about amoebas, train tracks, child’s drawings, shtukas, and bubble trees? Not to mention perversity and cacti.

Posted by: John Baez on January 21, 2007 2:45 AM | Permalink | Reply to this

Re: Terminology

Hmm. I guess things have moved on since the times when people wouldn’t stray far from terms like ‘normal’, ‘regular’ and ‘weak’.

I’d still like to hear of Janusian objects sitting in a pair of bicategories if such a thing exists.

Posted by: David Corfield on January 21, 2007 5:03 PM | Permalink | Reply to this

Re: Terminology

I’ve never seen the words ‘Janusian’ or ‘Janus-like’; what I’ve seen is ‘Janus-faced’. This is the one that gets the most Google hits, and it has the advantage of suggesting the word ‘two-faced’, to help out people who don’t know their Roman mythology.

Unfortunately, ‘Janus-faced’, like ‘two-faced’, has the connotation of ‘deceitful’. But, I guess ‘schizophrenic’ is also negative. I’d suggest ‘ambidextrous’, but that’s already taken.

Posted by: John Baez on January 21, 2007 2:31 AM | Permalink | Reply to this

Re: Terminology

How about ‘Janusian objects’ after the Roman God Janus?

Ooh, I like that too! And you can justify it to those conservative mathematicians using Rothenberg’s interpretation, since even conservative mathematicians love to think of themselves as creative.

Posted by: Toby Bartels on January 21, 2007 4:13 AM | Permalink | Reply to this

Re: Terminology

Janusian thinking–“actively conceiving two or more opposite or antithetical ideas, concepts, or images simultaneously,” according to the author’s definition–is proposed as a specific thought process that operates in the act of creation.

If there is some residual sense of “oppositeness” to “Janusian” or “Janus-faced”, as in the figure of Janus facing in opposite directions, then it doesn’t seem to me all that accurate for describing objects formerly known as schizophrenic.

“Schizophrenic”, literally “split-minded”, was perhaps ill-conceived for the reason Tom gives, but perhaps if we just keep the “schizo” and change the “phrenic” to something more accurate? I thought of “schizomorphic”, which already sounds mathematical, but there may be better options. (“Schizomorphic” already has various technical meanings, including an Aristasian one, but I think these could safely be ignored.)

Or, how about “ambimorphic”? I think I like that even more.

Posted by: Todd Trimble on January 21, 2007 9:50 AM | Permalink | Reply to this

Re: Terminology

I feel a little silly: “Janusian” is certainly appropriate (thinking of the object as a target object in a category and in its opposite). I was somehow getting diverted by a different sense of “opposite” – sorry!

I’m not prepared to do a complete about-face just yet – for some reason, “Janusian” sounds to me just a little precious, or like a word Gene Roddenberry might have used. I may be in the minority though.

Posted by: Todd Trimble on January 21, 2007 2:47 PM | Permalink | Reply to this

Re: Terminology

I agree that something doesn’t quite sit right with me about the term. On the other hand, if we end up deciding mythology is fair game I’ll work on finding something about tangles to name after Shelob.

Posted by: John Armstrong on January 21, 2007 4:05 PM | Permalink | Reply to this

Janice Ian, musician; Re: Terminology

“Janusian” was what I thought I heard when, in the 1960s, I heard the name of the singer-songwriter Janice Ian. See her home page
www.janisian.com/

Then I heard and liked her song “at seventeen” – which I already knew was a Fermat Prime. Finally, I got to meet her, at a World Science Fiction convention, as an anthology of stories was written based on her lyrics.

So I have no problem with the term. Is cerberus-headed the phrase from triality? Or tridental?

Posted by: Jonathan Vos Post on January 23, 2007 7:41 AM | Permalink | Reply to this

Re: Terminology

The reference to Janus in a similar (but informal) context goes back to Weyl in his 1932 paper “Topology and Abstract Algebra as Two Roads of Mathematical Comprehension” (1995 English translation available here: http://www.jstor.org/stable/2975040). There he speaks of the reals as “a Janus head with two oppositely directed faces”, referring to it being the domain of algebraic operations but also a continuous manifold, and to the interplay between the two structures.

Posted by: rsb on August 3, 2013 2:53 AM | Permalink | Reply to this

Re: Terminology

The reference to Janus in a similar (but informal) context goes back to Weyl in his 1932 paper “Topology and Abstract Algebra as Two Roads of Mathematical Comprehension” (1995 English translation available here: http://www.jstor.org/stable/2975040). There he speaks of the reals as “a Janus head with two oppositely directed faces”, referring to it being the domain of algebraic operations but also a continuous manifold, and to the interplay between the two structures.

Posted by: rsb on August 3, 2013 2:53 AM | Permalink | Reply to this

Re: Terminology

According to the OED, the word “schizophrenia” entered the English language in 1910, as an invented name for the mental illness. This illness has nothing to do with “split personalities”. Unfortunately, well-educated people at the time had a better knowledge of Greek than of psychiatry, and so the literal meaning “split mind” subsequently crept into the language. The OED cites TS Eliot as misusing the term in 1933 (Use of Poetry and Use of Criticism). So while I agree that we should now undo this ignorance, I wanted to point out that Simmons etc do have admirable company in their mistake.

Posted by: Sam on October 30, 2010 9:07 AM | Permalink | Reply to this

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