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February 4, 2008

Albert Lautman

Posted by David Corfield

Sifting through some old papers yesterday, I was reminded of my first forays into philosophy of mathematics while living in Paris. One very important influence on me was Albert Lautman (1908-1944). (The French Wikipedia has a little more on him than the English.) He’s often paired with Jean Cavaillès, understandably as they were friends, philosophers of mathematics, and both died as members of the French Resistance. Somehow I never got on with Cavaillès, even though he’s found greater favour in the Anglophone world. In Lautman, on the other hand, I recognised the kind of aesthetic sensibility I was later to enjoy in Mac Lane’s Mathematics: Form and Function and in This Week’s Finds.

I haven’t read Lautman in around 18 years, aside from a stolen hour spent in the Bodleian in Oxford, so went to see what was available on the Web. Not much it appears, but there is at least a translation of the introduction to Essai sur les notions de structure et d’existence en mathématiques: Les schémas de structure. Here are a couple of excerpts:

The structural design and the dynamic design of mathematics first of all seem to oppose themselves: the one indeed tends to regard a mathematical theory as a completed whole, independent of time, the other on the contrary does not separate the temporal stages from its development; for the former, the theories are like beings qualitatively distinct from each other, while the latter sees in each one an infinite power of expansion beyond its limits and connection with the others, because it affirms itself as the unity of the intelligence. We would however like, in the following pages, to try to develop a design of mathematical reality where the fixity of logical notions is combined with the movement that lives through these theories.

A dialectical action is constantly played out in the background and it is in order to clarify this that our six chapters will converge on this point. The first three chapters deal especially with de-structured mathematical notions. We study in chapter I (the local and the global) the almost organic solidarity which pushes the parts to be organized in a whole and the whole to be reflected in them; we examine then in chapter II (Intrinsic properties and inductive properties) if it is possible to bring back for the relations that a mathematical being supports with an ambient milieu, in the characteristic inherent properties of this being. We show in chapter III (rise towards completion) how the structure of an imperfect being can sometimes preform the existence of a perfect being in which any imperfection has disappeared. Then the three chapters relating to the concept of existence come. We try to develop in chapter IV (Essence and Existence) a new theory of the relations of essence and existence where one sees the structure of a being to be interpreted in terms of existence for beings other than the being of which one studies the structure. Chapter V (the Mixed) describes certain intermediate Mixtures between different kinds of Beings and whose consideration is often necessary to operate the passage of one kind of being to another kind of being; our final chapter (Of the exceptional character of existence) finally describes the processes by which a being can be distinguished within an infinity from the others.

What I remember, and I will have surely got some of the details wrong, of that third chapter ‘rise towards completion’ is a Galoisian comparison of covering spaces and field extensions, the ridding of the ‘imperfections’ of nontrivial first homotopy and algebraic nonclosure, respectively.

But the really bold part of Lautman’s chapter was to liken this situation to the one Descartes finds himself in when, having established that he exists, he wishes then to establish the existence of a Perfect Being, or God. Descartes argues that he recognises that he is imperfect, but that he has an idea of perfection. This idea must have arisen in him through a perfect Being, as one might say that the failure of factorisability of polynomials in \mathbb{Q} prefigure its algebraic completion. There may have been an additional parallel between intermediate fields and angelic beings who lack some of our imperfections and some of God’s perfections.

Even if you find this philosophical analogy too fanciful for you taste, Lautman clearly reveal mathematical perspicacity. I remember that Jean Dieudonné in a preface to one of Lautman’s books praised him for his perception of structural similarity across fields, and claimed that he’d just about discovered the concept of a representable functor (many years, note, before category theory saw the light of day).

As a small act of homage, and because I’d really like to discover the answer, let’s see what we know about about taking Galois up the nn-category ladder. So how does the deck transformation-simply connected covering space story run for higher homotopy? E.g., if we take CP CP^{\infty}, as K(,2)K(\mathbb{Z}, 2), is there a deck transformation story, featuring a covering space with trivial second homotopy?

What’s the closest thing we have to 2-Galois for factorising polynomials? Is it needed for species/structure types?

Of course, we’d like a very general account. As we suspect that practitioners of modal logic have already stumbled on categorified predicate logic, we need a 2-Galois theory to help us categorify what Todd told us about here. In particular,

any relation RA×BR \subseteq A \times B whatsoever can be used to set up a Galois connection between subsets AA' of AA and subsets BB' of BB: A{aA: b:B(a,b)R}B{bB: a:A(a,b)R}iff\frac{A' \subseteq \{a \in A: \forall_{b: B'} (a, b) \in R\} }{B' \subseteq \{b \in B: \forall_{a: A'} (a, b) \in R\} } iff Indeed, each of the conditions above and below the bar is equivalent to the condition A×BRA' \times B' \subseteq R.

Can that be so hard to categorify? Just think, if we get it right then we’ll be able to work out a categorifed cartesian theology!

Posted at February 4, 2008 2:27 PM UTC

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Re: Albert Lautman

To make a start on categorifying what Todd said, we could note that a relation between AA and BB gives rise to order preserving maps between P(A)P(A) and P(B) opP(B)^{op} and vice versa which are adjoint.

Now you might hope that a permutation representation of the category C×D opC \times D^{op} would similarly give rise to adjoint functors between the categories CC-Set and (DSet) op(D-Set)^{op}.

Restricting to groups, does a representation of G×HG \times H allow such mappings between GG-Set and (HSet) op(H-Set)^{op}?

Posted by: David Corfield on February 5, 2008 9:01 AM | Permalink | Reply to this

Re: Albert Lautman

That’s essentially right, and it’s an interesting observation which I think is due to Lawvere (who else?). There are some slightly persnickety (co, contra) variances to keep track of, but let’s have a look.

Categorifying, one naturally replaces the power set PAP A in the 1-topos SetSet by Set C opSet^{C^{op}} in the 2-topos CatCat, and the singleton mapping {}:APA\{- \}: A \to P A by the Yoneda embedding y C:CSet C opy_C: C \to Set^{C^{op}}. Here, Set C opSet^{C^{op}} is the category of right (i.e., contravariant) CC-sets.

In the categorical literature (e.g., in the work of Street and Walters on yoneda structures, and more recently in Mark Weber’s work on 2-toposes), it is common practice to abbreviate Set C opSet^{C^{op}} to PCP C, and I’ll do that here as well.

Some general principles.

It is pertinent to remark that PCP C is the free cocompletion of CC [assuming CC is small]: if EE is cocomplete, then a functor F:CEF: C \to E determines, uniquely up to isomorphism, a cocontinuous functor

F^:PCE\hat{F}: P C \to E

which extends FF along the Yoneda embedding y C:CPCy_C: C \to P C. The way this works is that F^\hat{F} assigns, to an object of PCP C, that is to say a weight X:C opSetX: C^{op} \to Set, the following weighted colimit in EE:

X CF= cX(c)F(c).X \otimes_C F = \int^c X(c) \cdot F(c).

By definition of weighted colimit, this fits in an adjunction formula

E(X CF,e)PC(X,hom E(F,e)).E(X \otimes_C F, e) \cong P C(X, hom_E(F-, e)).

It follows that in fact F^\hat{F} is not just cocontinuous, but actually a left adjoint: its right adjoint is the functor EPCE \to P C which takes an object ee to the functor hom E(F,e):C opSethom_E(F-, e): C^{op} \to Set.

Dually, (PD) op(P D)^{op} is the free completion of DD. Now, this category is not only complete, but cocomplete (since PDP D, being a power of SetSet, is complete). Therefore, by the free cocompletion property above, a functor

R:C(PD) op=(Set op) DR: C \to (P D)^{op} = (Set^{op})^D

induces an adjoint pair whose left adjoint is R^\hat{R}, and whose right adjoint takes an object WW of (PD) op(P D)^{op} to the functor

C opSet:chom (PD) op(Rc,W)=PD(W,Rc).C^{op} \to Set: c \mapsto hom_{(P D)^{op}}(R c, W) = P D(W-, R c).

The main point.

Let’s recap. Start with a 2-relation

R:C op×D opSet.R: C^{op} \times D^{op} \to Set.

This induces, in the obvious way, a functor

R˜:C(Set op) D=(PD) op\tilde{R}: C \to (Set^{op})^D = (P D)^{op}

and thence an adjunction whose right adjoint part is a functor (PD) opPC(P D)^{op} \to P C. Our calculation above shows that this is precisely the contravariant functor PDPCP D \to P C which takes a weight WW in PDP D to

chom PD(W,R(c,)).c \mapsto hom_{P D}(W, R(c, -)).

Does this look familiar yet? In fact, all we are really saying is that given a weight XX in PCP C, we have a natural bijective correspondence (using ? and - to denote dummy variables cc, dd):

X(?)hom PD(W(),R(?,))X(?)×W()R(?,).\frac{X(?) \to hom_{P D}(W(-), R(?, -))}{X(?) \times W(-) \to R(?, -)}.

We can turn this around, by a kind of reciprocity familiar from Galois connections, into the form

X(?)×W()R(?,)W()hom PC(X(?),R(?,))\frac{X(?) \times W(-) \to R(?, -)}{W(-) \to hom_{P C}(X(?), R(?, -))}

Now we arrive at the categorification of the Galois connection which David referred to (coming from my post on Concrete Groups and Axiomatic Theories): each 2-relation R:C op×D opSetR: C^{op} \times D^{op} \to Set induces a pair of contravariant functors

PCPD:Xhom PC(X(?),R(?,))P C \to P D: X \mapsto hom_{P C}(X(?), R(?, -))

PDPC:Whom PD(W(),R(?,))P D \to P C: W \mapsto hom_{P D}(W(-), R(?, -))

which fit into an adjunction

(PC(PD) op)((PD) opPC).(P C \to (P D)^{op}) \dashv ((P D)^{op} \to P C).

At this point, we can if we like eliminate all the ‘ops’, and say: a 2-relation R:C×DSetR: C \times D \to Set induces a contravariant adjunction between Set CSet^C and Set DSet^D. This implies, as a particular case, what David said:

Restricting to groups, a representation of G×HG \times H allows such a mapping [an adjunction] between GG-SetSet and (HH-Set) opSet)^{op}.

Some loose ends.

I’d also like to mention, for those following the thread on Comparative Smootheology, that this categorified Galois connection is what Andrew Stacey was working out, where he describes a process which passes from a presheaf to a copresheaf, and vice-versa. It is the particular case where the 2-relation RR is the hom-functor S op×SSetS^{op} \times S \to Set; the corresponding adjunction is something which Lawvere calls conjugation.

Finally, a small comment on categorifying first-order logical formulae. For the ordinary Galois connection induced by a relation RA×BR \subseteq A \times B, I had written down an equivalent pair of formulae:

A{aA: b:B(a,b)R}B{bB: aA(a,b)R}iff\frac{A′ \subseteq \{a \in A: \forall_{b: B'} (a,b) \in R\}}{B′ \subseteq \{b \in B: \forall_{a \in A'} (a,b) \in R\}}iff

For purposes of categorification, it is better if we think of a subset AA' of AA as a predicate on AA, i.e., as a truth-valued function A():A2A'(-): A \to \mathbf{2}. Then, we would write

A(a)( b:BB(b)R(a,b))B(b)( a:AA(a)R(a,b))iff\frac{A'(a) \leq (\forall_{b: B} B'(b) \Rightarrow R(a, b))}{B'(b) \leq (\forall_{a: A} A'(a) \Rightarrow R(a, b))}iff

where \Rightarrow is the implication operator (internal hom) in 2\mathbf{2}.

Categorifying this equivalence, we have an adjunction [a natural bijection]

A(a) b:Bhom(B(b),R(a,b))B(b) a:Ahom(A(a),R(a,b))\frac{A'(a) \to \int_{b: B} hom(B'(b), R(a, b))}{B'(b) \to \int_{a: A} hom(A'(a), R(a, b))}

where now AA', BB' are weights, and the categorified form of universal quantification is given by taking ends. These end formulas give the correct SetSet-valued homs on presheaf categories PAP A, PBP B, in exact accordance with our calculations above, where we had

A(?)hom PB(B(),R(?,))B()hom PA(A(?),R(?,)).\frac{A'(?) \to hom_{P B}(B'(-), R(?, -))}{B'(-) \to hom_{P A}(A'(?), R(?, -))}.

Posted by: Todd Trimble on February 8, 2008 6:12 AM | Permalink | Reply to this

Re: Albert Lautman

Great, thanks! I’ll go and study this in search of an inspired guess for candidates for a categorified version of your relation between theories and symmetry groups, though I rather suspect someone has done more than guess already:

Th(G,X)(n)={pP(X n): g:Gp(gx 1,,gx n)=p(x 1,,x n)}Th(G, X)(n) = \{p \in P(X^n): \forall_{g: G} p(g x_1, \ldots, g x_n) = p(x_1, \ldots, x_n)\} and Aut T(X)={gX!: p:Tp(gx 1,,gx n)=p(x 1,,x n)}.Aut_T(X) = \{g \in X!: \forall_{p: T} p(g x_1, \ldots, g x_n) = p(x_1, \ldots, x_n)\}.

I’d like to know if

According to Tarski’s thesis, a notion is “logical” if it is preserved by an arbitrary automorphism,

what will be preserved by arbitrary 2-automorphism.

I supposed I shouldn’t be surprised, in view of the nature of this blog, but your saying

I’d also like to mention…that this categorified Galois connection is what Andrew Stacey was working out…

makes me realise how much of our effort is just chipping away at the same block.

Posted by: David Corfield on February 8, 2008 11:41 AM | Permalink | Reply to this

Re: Albert Lautman

I guess we ought to try to understand other attempts at 2-Galois. Pierre Cartier tells us (p. 404) the story of the fundamental group of a space XX, based at a point aa, as the symmetry group of that point.

If we take the category (topos, in fact) of locally constant sheaves over XX, then the point aa in XX gives us a functor to Set, which reads off for any sheaf the fibre at that point.

The automorphism group of this functor, i.e., the invertible natural transformations, were defined by Grothendieck as π 1(X;a)\pi_1(X; a).

Presumably it wouldn’t be hard to extend this to the fundamental groupoid with objects as fibre functors for different points of XX.

Now has anyone other than Bertrand Toen tried to take things a stage further? Not that there’s anything wrong with Toward a Galoisian interpretation of homotopy theory, but it does try to do the whole thing at once by capturing all of XX’s homotopy via \infty-stacks, and I’d just like to the thing worked out for second homotopy groups in a concrete-ish case.

What might happen? You look for a 2-category of gerbes on XX, take its fibre 2-functor at a point to Groupoid, then look for symmetries of the identity natural transformation on that 2-functor?

Posted by: David Corfield on February 7, 2008 1:03 PM | Permalink | Reply to this

Re: Albert Lautman

For the 2-categorical Galois correspondance, there is this paper of I. Waschkies and P. Polesello

http://www.intlpress.com/hha/v7/n1/a7/pdf

Posted by: Denis-Charles Cisinski on February 8, 2008 11:01 AM | Permalink | Reply to this

Re: Albert Lautman

Thanks! This is just the sort of thing I was after. Now to see how it relates to what Todd was telling me.

Posted by: David Corfield on February 8, 2008 11:45 AM | Permalink | Reply to this
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