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December 8, 2009

When Naturality Fails

Posted by David Corfield

I started a page at nnLab on the notion of injective hulls. It seems to cover a range of interesting constructions, including algebraic field extensions and Dedekind-MacNeille completion. But it appears that it is not ‘natural’. So I asked at the nnForum

Does this tell us something interesting?

Toby replied

This has always been a strange thing to beginning category theorists (and maybe still for the rest of us). The usual example seems to be the algebraic closure of a field (although that’s an injective hull only in a slightly unobvious category, now noted on our page). Any two algebraic closures are isomorphic, but not canonically so, and this applies to injective hulls in general.

It should tell us something, but I don’t know what.

Can anyone help us?

Posted at December 8, 2009 9:43 AM UTC

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Re: When Naturality Fails

The algebraic closure, the MacNeille completion, but also the Hahn-Banach extension (Joy of Cats 9.3.5), all live in a topos different from the one the orginal object lives in.

Maybe this provides a clue.

Posted by: Bas Spitters on December 8, 2009 11:53 AM | Permalink | Reply to this

Re: When Naturality Fails

Looks interesting. Could you tell us a little more about this please?

Posted by: David Corfield on December 8, 2009 3:49 PM | Permalink | Reply to this

Re: When Naturality Fails

Is this another way of saying that the right thing to consider is the groupoid of injective hulls, rather than one particular? I seem to recall an algebra professor painting exactly this sort of picture to rederive Galois theory.

Posted by: some guy on the street on December 8, 2009 4:44 PM | Permalink | Reply to this

Re: When Naturality Fails

This is essentially the view I would have taken based on the only case I’ve thought about. For a field FF of characteristic zero, an algebraic closure

FF¯F\hookrightarrow \bar{F}

is the same as a geometric point

Spec(F¯)Spec(F).Spec(\bar{F})\rightarrow Spec(F).

So one wouldn’t expect it to be any more functorial than choosing a point on a space.

Posted by: Minhyong Kim on December 12, 2009 1:01 AM | Permalink | Reply to this

Re: When Naturality Fails

The MacNeille completion of the rationals coincide with the Dedekind real numbers in the Gleason cover of the topos.

The Hahn-Banach theorem states that:
If A subspace B, then every
element of Fn1(A) extends Fn1(B).
It can be proved localically as the existence of an epi from Fn1(B) to Fn1(A). Hence the Hahn-Banach extention exists in sheaves over Fn1(B).

For the algebraic closure see:
This paper by Coquand and the references [8,12] at the end.

Posted by: Bas Spitters on December 10, 2009 11:32 AM | Permalink | Reply to this

Re: When Naturality Fails

This paper by Coquand

Broken link.

Posted by: Toby Bartels on December 10, 2009 8:57 PM | Permalink | Reply to this

Re: When Naturality Fails

Bas Spitters hits the nail on the head. For a topos E one can talk of an E-geometric theory T, whose models live in E-toposes. The yoga of algebraic theories, the existence of free functors left adjoint to forgetful functors induced by maps of theories etc, lifts I believe to geometric theories, but stepping up one degree in the categorical hierarchy. The classic example is Monique Hakim’s thesis. This says that given a commutative ring A in a topos E, the free local ring on A lives in the E-topos Spec(A). Of course you have to formulate the E-geometric theory of local rings over A, describe the map of geometric theories from rings to it, etc. This stuff has been around a while. Has nobody’s student been tasked with writing it all down prettily yet?

Posted by: Gavin Wraith on December 8, 2009 9:08 PM | Permalink | Reply to this

Re: When Naturality Fails

I don’t understand what Bas and Gavin are getting at. If I have a field in the topos SetSet, then surely its algebraic closure is also a field in SetSet. Are you saying that rather than asking for “the free algebraically closed field containing kk” one should ask instead for “the free topos containing an algebraically closed field containing kk”?

Posted by: Mike Shulman on December 8, 2009 10:26 PM | Permalink | PGP Sig | Reply to this

Re: When Naturality Fails

Well maybe this is a stupid suggestion, but surely the (separable?) algebraic closure k¯\overline{k} of kk sits in the topos of Gal(k¯/k)Gal(\overline{k}/k)-sets?

Posted by: David Roberts on December 8, 2009 10:55 PM | Permalink | Reply to this

Re: When Naturality Fails

Let us look at a simple case - the algebraic closure of the field of real numbers. One way of looking at the field of complex numbers is as a field in the topos of sets-with-involution, complex conjugation being the involution in this case. In Topos Theoretic Methods in Geometry, Aarhus University Various Pub. Series No. 30, May 1979, I showed with little pictures how you adjoin freely square roots of -1 to any field in a topos. What you get does not live in the same topos.
If you take a larger view of all this, what you are doing is using 2-pullbacks in the 2-category of toposes, geometric morphisms and natural maps.

Posted by: Gavin Wraith on December 9, 2009 9:58 AM | Permalink | Reply to this

Re: When Naturality Fails

What you’re saying still sounds cryptic to me, but I’m guessing that you’re saying the answer to my question is “yes”? In that case, what is the explanation for why the complex numbers are generally considered to live in SetSet rather than some other topos?

Posted by: Mike Shulman on December 10, 2009 8:51 AM | Permalink | PGP Sig | Reply to this

Re: When Naturality Fails

Since Gavin’s comment sounds ‘cryptic’ to you, let me give my naive oversimplified interpretation of what he was saying. Perhaps this part was the part that made sense already, but anyway:

If you adjoin a square root of 1-1 to a field that lacks one, you’re really adjoining two square roots of 1-1, related by a symmetry: ‘complex conjugation’, in the case we know and love. And it’s probably better to think of it like this: you adjoin two square roots of 1-1, without any way to tell which one is which.

So, this process really takes a field without a square root of 1-1 living in the topos of sets to a field with two square roots of 1-1 living in the topos of sets with /2\mathbb{Z}/2 action.

And of course the same story applies to other more fancy Galois extensions.

In that case, what is the explanation for why the complex numbers are generally considered to live in Set rather than some other topos?

Well, that’s easy. It’s because most people don’t know about topoi: they think math is about sets!

By the way: did you know electrical engineers use jj instead of ii to denote the square root of 1-1? And do you know why? And did you know that actually j=ij = -i?

Posted by: John Baez on December 14, 2009 4:39 AM | Permalink | Reply to this

Re: When Naturality Fails

JB said:

did you know electrical engineers use jj instead of ii to denote the square root of 1-1?

Yes.

And do you know why?

I heard it was because II denotes current, and there was a concern there might be confusion. Which is ironic, if true, because, j\mathbf{j} denotes current density (and jj denotes 44-current density!)

And did you know that actually j=ij=-i?

Well, it’s as good a story as any …

Posted by: Tim Silverman on December 14, 2009 4:23 PM | Permalink | Reply to this

Re: When Naturality Fails

That makes sense, and it’s what I was guessing he was getting at. My question was then, is Set /2Set^{\mathbb{Z}/2} equipped with the complex numbers the “free topos containing a field that extends \mathbb{R} with square roots of 1-1?”

In that case, what is the explanation for why the complex numbers are generally considered to live in Set rather than some other topos?

Well, that’s easy. It’s because most people don’t know about topoi: they think math is about sets!

Well, yes, but what I meant to ask was, how do we recover the usual complex numbers from these fancy complex numbers in Set /2Set^{\mathbb{Z}/2}? If the fancy ones have the universal property I suggested above, then there should be a map Set /2SetSet^{\mathbb{Z}/2} \to Set of some sort taking the fancy ones to the usual ones? I’m guessing this would actually be a geometric morphism SetSet /2Set \to Set^{\mathbb{Z}/2}, since it’s the inverse image parts of geometric morphisms that carry “algebro-logical” data. And indeed, it does seem that the obvious geometric morphism SetSet /2Set \to Set^{\mathbb{Z}/2}, whose inverse image functor just forgets the /2\mathbb{Z}/2 action, has this property. Is that the point?

By the way: did you know electrical engineers use jj instead of ii to denote the square root of 1-1? And do you know why?

I seem to remember encountering that before. Is it because they use ii for current?

And did you know that actually j=ij=-i?

I don’t even know what that statement means. Are you saying that they measure angles in the complex plane clockwise instead of counterclockwise?

Posted by: Mike Shulman on December 14, 2009 4:28 PM | Permalink | PGP Sig | Reply to this

Re: When Naturality Fails

And did you know that actually j=ij=−i?

I don’t even know what that statement means. Are you saying that they measure angles in the complex plane clockwise instead of counterclockwise?

What are these “clockwise” and “counterclockwise” that you write of?

Actually, it’s a lie! jij\neq -i! The root jj belongs to the extension field [j]/(j 2+1)\mathbb{R}[j]/(j^2+1) and ii belongs to the extension field [i]/(i 2+1)\mathbb{R}[i]/(i^2+1). So they turn out to be isomorphic; isomorphic in two ways, even. There’s one isomorphism generated by iji\mapsto j and another generated by iji\mapsto -j. Now, maybe, by jj you really mean i-i; but that just confuses me, and sounds “evil”.

Posted by: Jesse McKeown on December 14, 2009 10:52 PM | Permalink | Reply to this

Re: When Naturality Fails

Precisely correct.

Posted by: Toby Bartels on December 15, 2009 12:09 AM | Permalink | Reply to this

Re: When Naturality Fails

Y’all, I think the j=ij=-i thing was a joke.

Posted by: James on December 15, 2009 2:06 AM | Permalink | Reply to this

Re: When Naturality Fails

Wait, is that a pun about “conventional” current vs. electron current? Hmmm???

Posted by: Jesse McKeown on December 14, 2009 11:32 PM | Permalink | Reply to this

Re: When Naturality Fails

If I may offer an opinion: “I” is the electrical current, so electrical engineers use “j” (they usually don’t care that j is used to denote a current density by physicists).
If you have a simple circuit with inductive reactance only the current will be lagging the voltage by 90 degrees, in a simple circuit with capacitive reactance only the current will be leading the voltage by 90 degrees, so however you define j, you will always get a positive reactance for one of the two circuits and a negative one for the other, so - just in case they use i = -j in order to do clockwise rotations instead of counterclockwise, which I was unaware of - what’s the point?

Hope I did not made a fool of myself.

Posted by: Tim vB on December 15, 2009 6:19 PM | Permalink | Reply to this

Re: When Naturality Fails

That just had to happen.
…i = -j in order to do counterclockwise rotations instead of clockwise…

Posted by: Tim vB on December 15, 2009 6:24 PM | Permalink | Reply to this

Re: When Naturality Fails

Well, yes, but the folks were clever, too, and defined the measures so that an inductance LL was like a resistance jLjL, while a capacitance CC is like a resistance 1jC\frac{1}{jC} — the generalized resistances being called impedances altogether. So (as Eric says below) in the frequency domain, Ohm’s law looks like v=iZ, v = iZ, so for the idealized circuits we get v=jiL v = jiL where voltage is a quarter phase in advance of the current, or else v=jiC v = -j\frac{i}{C} giving a phase-delayed voltage.

Of course, when I see something like v=jiLv=jiL, I’m tempted to simplify it to v=kLv=-kL… :-P

Posted by: some guy on the street on December 19, 2009 2:44 PM | Permalink | Reply to this

Re: When Naturality Fails

Like the silly fellow I am, I rather missed your question

what’s the point?

to which I can only respond “there can’t be such a point, because it won’t work”. I’m unaware of any method for defining “clockwise” that doesn’t hinge on neutrinos all having the same handedness (or anitneutrinos, depending). And if the Universe is big enough and weird enough, I can imagine even that definition suffering from historico-geographical bias!

I want to toss off the logician’s word “indiscernible” in reference to the set {i,i}\{i,-i\}, only I’m not sure if I’m using it right.

Posted by: some guy on the street on December 19, 2009 2:54 PM | Permalink | Reply to this

Re: When Naturality Fails

My claim that j=ij = -i was a joke, meant to illustrate the basic point of Galois theory: there’s no way to tell.

My claim that engineers use the letter jj to mean the square root of 1-1 was not a joke.

The question is: why?

I’ve always heard electrical engineers use jj to mean the square root of 1-1 because II means ‘current’.

Does this mean they can’t tell uppercase letters from lowercase letters? That would make a great wisecrack — but a great way to make engineers even more annoyed at mathematicians and their superior attitudes. But it doesn’t seem to be true, at least not now. Plenty of engineers do use II for electric current and JJ for current density, these being related by

J=IAJ = I A

where AA is the cross-sectional area of the wire through which the current is flowing. So, they’re equally screwed regardless of whether they use ii or jj to mean the square root of 1-1. For a nice explanation, see Glenn Elert’s page on electric current.

But here’s the big question: what fool dreamt up the idea of using II to mean ‘current’ in the first place???

Apparently it was Michael Faraday, who was far from an fool. Glenn Elert writes:

Why II for current? Why not cc for current or ff for flow?

The word “ion” is ancient Greek for “going” and was coined by Michael Faraday (1791-1867) England to designate those electrically charged particles that migrate to one or another pole when an electric field is set up in a solution. The ion that moves “uphill” (e.g., chlorine, nitrogen) to the positive electrode he called anion (ανα = up) and the ion that moves “downhill” (e.g., hydrogen, zinc) the cation (κατα = down). He called the locations where the electrically charged ions “exit” electrodes (οδος = path, street, way; compare to εξοδος = way out, exit, exodus).

And here’s something from Michael Faraday, His Life and Work by Silvanus P. Thompson — the electrical engineer who wrote the world’s best calculus book! (I learned calculus out of it!) As quoted by Glenn Elert, Thompson wrote:

The ion of physics is from Greek ιον, present participle of ιεναι, to go. Joseph E. Shipley, Dictionary of Word Origins, 1945.

Greek ιον, something that goes, neuter present participle of ιεναι, to go. American Heritage Dictionary, 2000.

The OED (1989, Second Edition) says [a. Gr. ιον, neut. pr. pple. of ιεναι to go.] Name given by Faraday to either of the constituents which pass to the “poles” or electrodes in electrolysis: the general term including anion and cation. In modern use, any individual atom, molecule, or group having a net electric charge (either positive or negative), whether in an electrolytic solution or not.

1834. William Whewell. Letter to Michael Faraday 5 May in I. Todhunter, William Whewell (1876) II. 182 For the two together you might use the term ions.

1834 Michael Faraday in Philosophical Transactions of the Royal Society. CXXIV. 79 Finally, I require a term to express those bodies which can pass to the electrodes … I propose to distinguish these bodies by calling those anions which go to the anode of the decomposing body; and those passing to the cathode, cations; and when I have occasion to speak of these together, I shall call them ions.

In the nineteenth century, geology, paleontology and physics enlarged their vocabularies with the help of William Whewell, a veritable mint of new coinages. He did not try for grand systems, however, but for small groups of linked words that would convey concepts without fixing theories. In response to Michael Faraday’s request for terms to describe his experiments in electrolysis, Whewell supplied anode, cathode, electrode and ion from non-committal Greek roots. J.L. Heilbron. “Coming to Terms.” Nature. Vol. 415, No. 585 (2002).

Forgotten Source — From Greek ion, literally “moving thing,”

From Aqua: Ion was the son of Creusa by either Apollo (the god of prophecy, the arts and archery) or Xuthus. There are different stories about his origin, but the one that’s most relevant to the meaning of his name goes like this: Apollo raped Creusa when she was already Xuthus’s wife, and Hermes (the messenger god) was appointed by Apollo to bring his son to the Oracle of Delphi to work under the priests for him. Many years later, Xuthus and Creusa consulted the Oracle because they couldn’t have children, and Xuthus was told that the first man that he met leaving the temple would be his son. Not surprisingly, it was Ion that he met first, and he gave him this name because he met him “on the way.”

And now Richard Garner has invented ionads, which generalize topological spaces. But the etymology here is quite different…

Posted by: John Baez on December 17, 2009 6:39 PM | Permalink | Reply to this

Re: When Naturality Fails

My claim that engineers use the letter jj to mean the square root of -1 was not a joke.

The question is: why?

I’ve always heard electrical engineers use jj to mean the square root of -1 because I means “current”.

Does this mean they can’t tell uppercase letters from lowercase letters?

As part physicist, part engineer, and (a small) part mathematician (who now works in finance!), I can tell you that only physicists use II for current :) Well, engineers may use II to mean current as a function of time.

However, engineers rarely study circuits as a function of time. They instead study circuits in the Fourier domain, i.e. as a function of (angular) frequency ω=2πf\omega = 2\pi f. Circuit quantities such as voltage and current in Fourier space are expressed in lower case. Therefore we have the duality

I(t)i(ω)I(t)\Leftrightarrow i(\omega)

and

V(t)v(ω).V(t)\Leftrightarrow v(\omega).

Then we go on to define “phasors” by

I(t)=ℝ𝕖[i(ω)e jωt]I(t) = \mathbb{Re}\left[i(\omega) e^{-j\omega t}\right]

and

V(t)=ℝ𝕖[v(ω)e jωt].V(t) = \mathbb{Re}\left[v(\omega) e^{-j\omega t}\right].

So the objects we work with are these “phasors” ii and vv and rarely ever bother with II and VV. Ohm’s law for engineers is

v=i𝒵,v = i \mathcal{Z},

where 𝒵\mathcal{Z} is a complex number called “impedance”.

If we used ii for a root of -1, then we’d have things like

Itiωi.\frac{\partial I}{\partial t}\Leftrightarrow i\omega i.

Instead, the convention is

Itjωi.\frac{\partial I}{\partial t}\Leftrightarrow -j\omega i.

So the reason is basically due to “current”, but more precisely “phasor current”, which is expressed as lower case ii.

PS: It felt good to use LaTeX after a hiatus for some reason :)

Posted by: Eric on December 18, 2009 1:45 AM | Permalink | Reply to this

Re: When Naturality Fails

Thanks for the explanation, Eric! I’ve been thinking about electrical circuits a lot lately, especially in the Fourier domain, but somehow I hadn’t noticed that electrical engineers use ii and vv as symbols for the Fourier transforms of II and VV.

I was really amused when I first heard about ‘phasors’ — it reminded me of Star Trek. I think most mathematicians have never heard of ‘phasors’.

Posted by: John Baez on December 22, 2009 9:04 AM | Permalink | Reply to this

Re: When Naturality Fails

I have read somewhere that someone (Faraday?) simply used the letters A-J for the quantities in EM. The names C and F have fallen out of favor, but the rest are still in Maxwells equation. Perhaps C and F meant charge density and voltage.

Posted by: Thomas Larsson on December 19, 2009 1:58 PM | Permalink | Reply to this

Re: When Naturality Fails

Is all of this related to the failure in some toposes of the theorem that, for RR a real-closed linearly ordered Heyting field, R[i]R[i] is algebraically closed? Either countable choice or excluded middle (or even something rather weaker) suffices to prove this, but you need something. (This is the algebraic half of the fundamental theorem of algebra, where the analytic half is the theorem that a located-Dedekind-complete linearly ordered Heyting field is real closed. That half is valid in the internal language of any topos with NNO.)

Posted by: Toby Bartels on December 14, 2009 5:35 AM | Permalink | Reply to this

Re: When Naturality Fails

Interesting question, Toby! This is the sort of thing some people on the category theory mailing list would know.

Posted by: John Baez on December 17, 2009 6:41 PM | Permalink | Reply to this

Re: When Naturality Fails

Without DC you cannot not prove that X^2-a has a root.
However, you can construct the multiset of all roots.
See this paper by Richman.

We have suggested that Richman actually constructs the formal space of all solutions. This gives the connection with the present discussion.

Posted by: Bas Spitters on January 13, 2010 1:18 PM | Permalink | Reply to this

Re: When Naturality Fails

Thanks for the link to your work. Do you have further references for the analogy between complete totally bounded metric spaces and compact overt locales? (top of page 9). That is something that I would like to understand better.

Incidentally, one does not need all of DC just to prove that x 2ax^2 - a always has a root in the complex numbers; as Richman's paper shows, a weak form of countable choice (which also follows from excluded middle) suffices for that case. (I mention this mostly because mathematicians using classical logic will not think that any choice at all is needed for this result.)

Posted by: Toby Bartels on January 13, 2010 1:48 PM | Permalink | Reply to this

Re: When Naturality Fails

I asked:

Do you have further references for the analogy between complete totally bounded metric spaces and compact overt locales?

Looks like Bas already wrote something about that in Locatedness and overt sublocales.

Posted by: Toby Bartels on January 29, 2010 6:18 PM | Permalink | Reply to this

Re: When Naturality Fails

Lubarsky and Richman just published a new paper on zero sets of polynomials.

Posted by: Bas Spitters on January 29, 2010 8:47 PM | Permalink | Reply to this

Re: When Naturality Fails

>Does this tell us something interesting?

I think it does. This is where higher dimensional categories and weak adjoints show up in the ordinary category theory! I attach as an explanation my message written as an answer for the discussion about this topic nine years ago. I did not get any reply on my entry those time but, in nine years there were a lot of progress. I hope it will be more understandable now.


To: categories@mta.ca
Subject: categories: Re: Functorial injective hull.
From: mbatanin@ics.mq.edu.au (Michael Batanin)
Date: Mon, 27 Mar 2000 18:24:57 +1100
Sender: cat-dist@mta.ca

This is just to give different point of view to the problem.

1. Suppose we have a category C and a full subcategory H (think of H as a
subcategory of injective objects). Suppose we have a functor E:C –> H
together with natural transformation
i: Id –> E(C)
such that i is monic and ,moreover, E is weak left adjoint to the inclusion
functor H –> C. Then I claim that under two additional conditions E is
genuine adjoint and i is unit of the adjunction. The conditions are:

a. the natural transformation i is identity on H.
b. E preseves monics.

The proof is just a repetition of P.Freyd proof. We have to prove that
the extension in the diagram

i:A –> E(A)
| /
f | /
| /
I

is unique for any morphism f and “injective” I.
By applying E to this diagram and using condition a) we see that it is
sufficient to prove that E(i) is identity. Repeating Freyd’s proof we see
that
E(i) is idempotent. Using condition b) we conclude that it is identity.

The conditions a) and b) are obviously satisfied in the case when E is
“injective hull functor” (of coarse a) is true up to iso, again see P,Freyd
proof).

As the unit of the adjunction is monic so the functor E reflects
epimorphisms. If in C we have that mono + epi implies iso we finally have
that i is iso and ,hence, the result of Adamek, Herrlish, Rosicky and
Tholen.

2. I would not dare to simply repeat P.Freyd argument if I don’t have another
proof (in a special but important case). The proof is much more
techniqual but I believe reflects another interesting side of the problem
of functoriality
of injective resolutions.

Consider the following bicategory.
The objects are categories enriched in the closed monoidal category of
(say bounded ) chain complexes.
The 1-arrows are enriched distributors.
The 2-arrows are homotopy classes of “coherent” natural transformations (i.e.
we localize the category of natural transformations with respect to the
class of morphisms which are level quaziisomorphisms).
We have to define the composition of 1-arrows. This requires some techniques
but in a few words the result is left derived functor of the composition of
enriched distributors.

The resulting bicategory is closed on the left and right. Now, for a chain
functor
K: A –> B
we can consider the right Kan extension of it along itself in the above
bicategory (codensity monad). The Kleisli category of it is called “strong
shape theory of K” Ssh_K. It is possible to prove, that:
c). If K is a full embedding and has an enriched left adjoint then Ssh_K
is just a Kleisli category of the corresponding monad on B.

d). If B is the category of chain complexes (say bounded)in an abelian
category with enough injectives and A is full subcategory of injective
chain complexes, then homology of Ssh_K(X,Y) are isomorphic to the right
derived functor of internal Hom of B. In particular, if X and Y are
concentrated in the dimension 0 the cohomology are just the correspobding
Ext’s.

Coming back to the original problem. If H\in C are abelian and satisfy the
conditions a),b) then, according to our calculations
the inclusion
K: Ch(H) —> Ch(C)
has a right adjoint and , hence,(by point c)
Ssh_K(X,Y) = Hom(X,E(Y))
for X,Y concentrated in dimension 0. Hence by d) the injective dimension of
C is 0 and we have the result again.

Another words the injective dimension is the obstruction for nonnaturality
of the injective hull.

The same sort of theory can be developed for simplicial (or Cat) enriched
situation see
Batanin.M, Categorical Strong Shape THeory, Cahiers de Topologie et Geom,
v,XXXVIII-1(1997)p. 3-66.

I think some other results of nonaturality (as , for example, the result of
Shakhmatov mentioned by AHRT on p.8) are related to these strong shape
categories.

Michael Batanin.

Posted by: Michael Batanin on December 10, 2009 3:32 AM | Permalink | Reply to this

Re: When Naturality Fails

Thanks to everyone for their rich comments. An overall sense of their direction still eludes me, however. And this is without including Todd Trimble’s bicategorical approach to MacNeille completion.

Posted by: David Corfield on December 12, 2009 11:25 AM | Permalink | Reply to this

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