background semi-algebraic sets

Schanuel and the Grothendieck (semi)ring of some theories

Tibor Beke

University of Massachusetts Lowell tibor [email protected]

Apr 17, 2019 I explicit finite presentation of the Grothendieck of the of semi-algebraically definable sets and functions over R.

background distributive categories semi-algebraic sets our hero

Stephen Schanuel (1934–2014), immortal for

I Schanuel’s lemma (homological algebra) I Schanuel’s conjecture (transcendental number theory) background distributive categories semi-algebraic sets semirings our hero

Stephen Schanuel (1934–2014), immortal for

I Schanuel’s lemma (homological algebra) I Schanuel’s conjecture (transcendental number theory) I explicit finite presentation of the Grothendieck semiring of the category of semi-algebraically definable sets and functions over R. background distributive categories semi-algebraic sets semirings same guy background distributive categories semi-algebraic sets semirings student feedback background distributive categories semi-algebraic sets semirings what has an associated Grothendieck (semi)ring?

Let X be a variety. Consider the free abelian group generated by classes of algebraic vector bundles over X and impose the relation [ξ] − [φ] + [χ] = 0 for each short exact sequence of vector bundles

0 → ξ → φ → χ → 0

The quotient is (the degree 0 part of) the algebraic K-theory of X . Multiplicative structure is induced by tensor product of bundles. background distributive categories semi-algebraic sets semirings what has an associated Grothendieck (semi)ring?

Let X be a (compact, Hausdorff) topological space. Consider the set of isomorphism classes of complex vector bundles over X and set [ξ] + [χ] := [ξ ⊕ χ] [ξ] · [χ] := [ξ ⊗ χ]

This is a semiring; the associated ring is (the degree 0 part of) the topological K-theory of X . background distributive categories semi-algebraic sets semirings Grothendieck’s Grothendieck ring of varieties

For any field k, let SK(vark ) be the semiring with generators [X ] for each variety X over k and relations

[X ] = [Y ] if X and Y are isomorphic over k

[X ] = [X − U] + [U] for every open subvariety U of X with complement X − U.

The product of [X ] and [Y ] is [X ⊗k Y ].

K(vark ) is the ring generated by the same generators and relations. I don’t think so but cases of interest to category theorists seem to be captured by

background distributive categories semi-algebraic sets semirings

More generally, “scissors-decomposition” type Grothendieck rings have been defined for I abelian categories (Grothendieck, ) I exact categories (Quillen) I triangulated categories (Verdier, Grothendieck) I categories with cofibrations (Waldhausen). And, one has the Grothendieck ring of a (model-theoretic) structure, and Grothendieck ring of a (first order) theory. Is there a common denominator for all these constructions? background distributive categories semi-algebraic sets semirings

More generally, “scissors-decomposition” type Grothendieck rings have been defined for I abelian categories (Grothendieck, ) I exact categories (Quillen) I triangulated categories (Verdier, Grothendieck) I categories with cofibrations (Waldhausen). And, one has the Grothendieck ring of a (model-theoretic) structure, and Grothendieck ring of a (first order) theory. Is there a common denominator for all these constructions? I don’t think so but cases of interest to category theorists seem to be captured by background distributive categories semi-algebraic sets semirings distributive category

Definition: Category C with finite products and such that the canonical maps

∅ → X × ∅

X × Y t X × Z → X × (Y t Z)

are (where ∅ is the initial object).

Remark Could play with two symmetric monoidal structures, one of which distributes across the other, but details are surprisingly involved. background distributive categories semi-algebraic sets semirings Grothendieck (semi)ring of a (small) distributive category C

SK(C) is the semiring whose elements are isomorphism classes [X ] of objects X , with [X ] · [Y ] := [X × Y ] and [X ] + [Y ] := [X t Y ].

K(C) is the abelian group generated by isomorphism classes [X ] of objects X , with the same relations.

Remark Schanuel calls SK(C) the “Burnside rig of C” in his pioneering article Negative sets have Euler characteristic and dimension. background distributive categories semi-algebraic sets semirings semiring

For the purposes of this talk ...

Definition A semiring is a set with two commutative, associative binary operations  and ~ with identity 0 resp. 1 such that

x ~ (y  z) = (x ~ y)  (x ~ z)

0 ~ x = 0 background distributive categories semi-algebraic sets semirings things I wish I knew # 1

Characterize the semirings that are isomorphic to the Grothendieck semiring of some small distributive category.

Asked by Schanuel in 1990 (cf. article quoted two slides ago).

He gives five necessary properties. Are they sufficient?

Some of those axioms are not equational (though they can be formalized in first-order geometric logic), so it’s not clear that free algebras exist in that variety of algebras. background distributive categories semi-algebraic sets semirings examples of semirings

(1) N with x ~ y := x · y and x  y := x + y is the free semiring on no generators.

(2) Let the underlying set be N ∪ {−∞} with operations  x y := x + y (x, y 6= −∞)  ~  x ~ −∞ := −∞ x y := max{x, y} (x, y 6= −∞)    x  −∞ := x

Same formulas will turn A t {−∞} into a semiring for any linearly ordered abelian monoid A. background distributive categories semi-algebraic sets semirings

N−∞ (a discrete variant of the tropical semiring)

N ∪ {−∞} with x ~ y := x + y and x  y := max{x, y} This is a finitely presentable semiring, isomorphic to

N[X ] modulo 1 + 1 = 1 and 1 + X = X under addition and multiplication of polynomials, sending −∞ 7→ 0 n 7→ X n

We will denote it N−∞h+, maxi or just N−∞. Aside: Same structure, written “multiplicatively”: formal symbol q (q > 1) underlying set := {0, 1, q, q2,..., qn,... }

x ~ y := x · y x  y := max{x, y} background distributive categories semi-algebraic sets semirings Grothendieck’s

Ring is the category of commutative, unital rings and homomorphisms. Adjunction

groth Ring  SemiRing inc For a semiring S, the elements of groth(S) are equivalence classes of pairs hx, yi, x, y ∈ S under the equivalence relation generated by

hx, yi ∼ hx  z, y  zi Natural semiring homomorphism

S → groth(S)

sending x to the class of hx, 0i. (2) groth N−∞h+, maxi = {0}

Canonical semiring homomorphism

S → groth(S) is injective iff S is additively cancellative, that is,

x  z = y  z implies x = y . Note that K(C) = groth SK(C) for any distributive category C.

background distributive categories semi-algebraic sets semirings examples

(1) groth Nh·, +i = Zh·, +i {0}

Canonical semiring homomorphism

S → groth(S) is injective iff S is additively cancellative, that is,

x  z = y  z implies x = y . Note that K(C) = groth SK(C) for any distributive category C.

background distributive categories semi-algebraic sets semirings examples

(1) groth Nh·, +i = Zh·, +i

(2) groth N−∞h+, maxi = Canonical semiring homomorphism

S → groth(S) is injective iff S is additively cancellative, that is,

x  z = y  z implies x = y . Note that K(C) = groth SK(C) for any distributive category C.

background distributive categories semi-algebraic sets semirings examples

(1) groth Nh·, +i = Zh·, +i

(2) groth N−∞h+, maxi = {0} background distributive categories semi-algebraic sets semirings examples

(1) groth Nh·, +i = Zh·, +i

(2) groth N−∞h+, maxi = {0}

Canonical semiring homomorphism

S → groth(S) is injective iff S is additively cancellative, that is,

x  z = y  z implies x = y . Note that K(C) = groth SK(C) for any distributive category C. background distributive categories semi-algebraic sets semirings model-theoretic geometry

n Set X (“points”) and for each n ∈ N, a set of subsets of X , n denoted Bn (“ definable subsets of X ”) such that n I Bn is closed under boolean operations in X I if U ∈ Bn and V ∈ Bm then U × V ∈ Bn+m n m I if U ∈ Bn then pr(U) ∈ Bm for projections pr : X → X I diagonals belong to Bn; singletons belong to B1.

See e.g. van den Dries: Tame geometry and o-minimal structures for a minimal set of axioms. Many-sorted version may be useful. background distributive categories semi-algebraic sets semirings distributive categories from model-theoretic geometries

Let X , Bn, n ∈ N form a geometry. Consider the category ( objects definable sets (i.e. elements of B ) Def(X):= n morphisms definable functions

i.e. a morphism from a definable U ⊆ X n to a definable V ⊆ X m is a function f : U → V whose graph belongs to Bn+m. background distributive categories semi-algebraic sets semirings syntactic categories

Let T be a theory in a logic (broadly interpreted). Consider the category ( objects hx, φ(x)i Def(T):= morphisms ψ(x, y) such that (see below)

where x is a tuple of variables and φ is formula with free variables among the x (considered up to renaming of variables). x, y are disjoint tuples of variables; ψ(x, y) is a morphism from hx, α(x)i to hy, β(y)i if T proves that ψ(x, y) is a function from α(x) to β(y). background distributive categories semi-algebraic sets semirings distributive categories from model-theoretic geometries

Proposition Def(X)

I has terminal object and pullbacks (so finite limits); they are computed as in Set I has finite coproducts I is distributive I is boolean (subobject lattices are boolean algebras; every subobject is a summand)

The nature of Def(T) depends on the fragment of logic underlying it. In the classical first order setting, it is a boolean category with finite limits, distributive (as soon as T proves the existence of two distinct singletons), but with no canonical underlying sets. background Schanuel’s presentation semi-algebraic sets our protagonists

‘Definable’ means ‘first order definable, with parameters’.  I SemiLinR is Def R; +, c · (−), < where c · (−) is multiplication by c (as unary function symbol)  I SemiAlgR is Def R; +, ×, <  I Constk is Def k; +, × (for some field k)

I BoundSemiLinR is the full subcategory of SemiLinR consisting n of bounded objects (for the usual metric on R ).

Note that BoundSemiLinR is a boolean (a fortiori, distributive) category, but does not coincide with Def(X) for any first order definable family over R (simply since the complement of a n bounded semi-linear subset of R is not bounded). I Unlike in the classical geometric case of “scissors congruence”, here we are responsible for a point for point between the unit cube and regular tetrahedron. I Dehn’s solution to Hilbert’s 3rd problem shows that the reassembly can’t be done using dissections and rigid motion (but certainly can if affine transformations are also allowed).

background Schanuel’s presentation semi-algebraic sets fun exercise

Show that the (closed) unit cube and (closed) regular tetrahedron

are isomorphic in SemiLinR. background Schanuel’s presentation semi-algebraic sets fun exercise

Show that the (closed) unit cube and (closed) regular tetrahedron

are isomorphic in SemiLinR.

I Unlike in the classical geometric case of “scissors congruence”, here we are responsible for a point for point bijection between the unit cube and regular tetrahedron. I Dehn’s solution to Hilbert’s 3rd problem shows that the reassembly can’t be done using dissections and rigid motion (but certainly can if affine transformations are also allowed). background Schanuel’s presentation semi-algebraic sets so pretty!

Theorem (Schanuel 1990)

SK(SemiAlgR) is a finitely presentable semiring, isomorphic to

N[X ] modulo X = 2X + 1 . background Schanuel’s presentation semi-algebraic sets step 1: algebraic games with semirings

There is a natural homomorphism

sch N[X ]/(X = 2X + 1) −−→ N−∞

since N−∞ has presentation N[X ]/(1 + 1 = 1; 1 + X = X ). The Grothendieck ring of N[X ]/(X = 2X + 1) is Z. The natural morphism

groth N[X ]/(X = 2X + 1) −−−→ Z simply evaluates a polynomial at X = −1. The product map

N[X ]/(X = 2X + 1) −→ N−∞ × Z is injective. Can be shown by degree-wise induction and playing with relations. background Schanuel’s presentation semi-algebraic sets step 2: existence of cell decompositions

There is a homomorphism

N[X ]/(X = 2X + 1) −→ SK(SemiAlgR) generated by X 7→ (0, 1).

Theorem The above map is surjective. This is deep. Needs the identification of first order definable with semi-algebraic sets (elimination of quantifiers, the Tarski-Seidenberg theorem) and the existence of open-cell decompositions (Hironaka: triangulations of compact semi-algebraic sets, Lojasiewicz: open cell decompositions of arbitrary semi-algebraic sets, Collins: cylindrical decompositions, extended by van den Dries to the o-minimal setting) background Schanuel’s presentation semi-algebraic sets step 3: existence of dimension

For X semi-algebraic,

dim(X ) := max n | there is a subspace of X n semi-algebraically homeomorphic to R defines a semiring homomorphism

dim : SK(SemiAlg) → N−∞

Remark: necessarily dim(∅) = −∞. background Schanuel’s presentation semi-algebraic sets step 4: combinatorial Euler characteristic

Let X be semi-algebraic and (V , S) an open-cell complex such that X is semi-algebraically homeomorphic to |S|.

P dim(U) Definition eu(X ) = U∈S (−1)

Theorem eu(X ) is independent of the open-cell decomposition chosen. The proof needs that any two semi-algebraic open-cell decompositions have a common semi-algebraic refinement. Proposition One has a semiring homomorphism

eu SK(SemiAlgR) −→ Z background Schanuel’s presentation semi-algebraic sets aside: the cohomological connection

∗ Let F be any field. Let H (−; F) denote sheaf (or equivalently, ∗ singular) cohomology and let Hc (−; F) denote cohomology with compact support. Let X be a semi-algebraic set. I If X is compact,

dim(X ) X i i eu(X ) = χ(X ) = (−1) dimF H (X ; F). i=0

I If X is locally compact,

dim(X ) X i i eu(X ) = χc (X ) = (−1) dimF Hc (X ; F). i=0 background Schanuel’s presentation semi-algebraic sets things I wish I knew # 2

I Is eu(X ) = χc (X ) for all semi-algebraic X , not just locally compact ones? Is there a cohomological interpretation of eu(X ) valid for all X ?

I χc is a proper (topological) homotopy invariant. Is that true for eu(X ) as well? background Schanuel’s presentation semi-algebraic sets

Theorem (TB, 2011) If X , Y are semi-algebraic (or more generally, o-minimal, belonging to possibly two distinct o-minimal structures) and topologically homeomorphic then eu(X ) = eu(Y ).

Proof reduces to locally compact case with the help of an intrinsically defined stratification of o-minimal sets.

Remark Already two polyhedra can be topologically homeomorphic but not semi-algebraically so (Milnor, counterexample to the polyhedral Hauptvermutung, 1961). background Schanuel’s presentation semi-algebraic sets Schanuel’s presentation

Commutative diagram of semigroup homomorphisms

sch×groth N[X ]/(X = 2X + 1) / / N−∞ × Z 7

X 7→(0,1) dim ×eu ) ) SK(SemiAlgR)

Down arrow is both injective and surjective, hence isomorphism. background Schanuel’s presentation semi-algebraic sets Corollary

(Schanuel 1990; o-minimal version: van den Dries, 1998)

If X and Y are semi-algebraic sets with dim(X ) = dim(Y ) and eu(X ) = eu(Y ) then they are equidecomposable (represent the

same element in SK(SemiAlgR). background Schanuel’s presentation semi-algebraic sets Proof

Induction on dimension. Valid for dim = 0. Suppose d = dim(X ) = dim(Y ) and eu(X ) = eu(Y ). ◦ Write ∆n for the open n-simplex. Have decompositions

d d G G ◦ G G ◦ X ! ∆i Y ! ∆i i=0 ni i=0 mi background Schanuel’s presentation semi-algebraic sets

Have decompositions

d d G G ◦ G G ◦ X ! ∆i Y ! ∆i i=0 ni i=0 mi Note ◦ ◦ ◦ ◦ ∆n ! ∆n t ∆n t ∆n−1

Can assume 0 < nd−1 and 0 < md−1 (apply relation if needed). ◦ Say 0 < nd 6 md . Use relation to replace the nd copies of ∆d in ◦ ◦ X by md copies of ∆d (affecting the number of ∆d−1 along the way). Once the same number of d-dimensional cells on both sides, remove them. Apply induction hypothesis. background Schanuel’s presentation semi-algebraic sets

Games with Schanuel’s diagram also prove: Theorem (Hironaka, Lojasiewicz, Schanuel) The inclusion of categories BoundedSemiLinR ,→ SemiAlgR induces an isomorphism

SK(BoundedSemiLinR) = SK(SemiAlgR)

Proposition The inclusion of categories SemiAlgQ ,→ SemiAlgR induces an isomorphism

SK(SemiAlgQ) = SK(SemiAlgR)

Corollary The Grothendieck semiring of the theory of real-closed fields is isomorphic to N[X ]/(X = 2X + 1) as well. background Schanuel’s presentation semi-algebraic sets from semiring to ring

K(SemiAlgR) = groth SK(SemiAlgR) is isomorphic to groth N[X ]/(X = 2X + 1) = Z.

Commutative diagram

dim ×eu SK(SemiAlg) / N−∞ × Z

groth groth  eu  K(SemiAlg) / Z

eu shows that K(SemiAlgR) −→ Z is an isomorphism (o-minimal version: van den Dries, 1998). background Schanuel’s presentation semi-algebraic sets

Theorem (Schanuel 1990)

SK(SemiLinR) is a finitely presentable semiring, isomorphic to

2 2 N[X , Y ]/(X = 2X + 1, Y = X + Y + 1, Y = 2Y + Y ).

Schanuel sketches a proof; full details seem to be unpublished. background Schanuel’s presentation semi-algebraic sets Schanuel’s functor

JoinSemiLatticeSemiRing: semiring hS, ~, i such that  is idempotent: x  x = x.

x 6 y ⇔ y = x  y

defines partial order on S, with respect to which ~ and  are monotone. x  y is the least upper bound of x, y. Remark For any JSLSemiRing S, one has groth(S) = {0}. Adjunction sch JSLSemiRing  SemiRing inc background Schanuel’s presentation semi-algebraic sets Schanuel’s functor

Adjunction sch JSLSemiRing  SemiRing inc For semiring hS, ~, i, define

x 4 y ⇔ ∃z ∈ S such that x  z = y  y  ...  y for some number of copies of y. 4 is preorder, compatible with ~ and . Define

x ∼ y iff x 4 and y 4 x.

S/ ∼ with operations induced by ~,  will be sch(S). background Schanuel’s presentation semi-algebraic sets universal mapping property

Let C be a distributive category and d : SK(C) → D any dimension morphism valued in an (additively) idempotent semiring D:

I d(∅) = 0 I d(∗) = 1 I d(A × B) = d(A) ~ d(B) I d(A t B) = d(A)  d(B) Then d factors uniquely as SK(C) → sch SK(C) → D. True, in particular, for any notion of dimension valued in a linearly ordered abelian monoid (with  as max). Dream Find a universal mapping property for Morley rank. background Schanuel’s presentation semi-algebraic sets things I wish I knew # 3

Can one understand distributive categories C whose SK(C) is finitely presentable (as semiring)?

Note that SK(C) being finitely presentable is a very strong notion of cellularity for C: the structure is generated (under multiplication and addition) by finitely many cells, subject to (the multiplicative-additive consequences of) finitely many ‘scissors’ or ‘decomposition-equivalence’ relations.

If SK(C) is finitely presentable, so are K(C) and sch SK(C), its Grothendieck ring and its ‘Schanuel dimension semilattice’. background Schanuel’s presentation semi-algebraic sets research exercise

Determine SK(−), K(−) and sch(−) for

I Constrk (algebraically constructible sets and maps over an algebraically closed field) I SemiAlg/X (overcategory) I SetsG (G-equivariant sets) I SemiAlg G (G-equivariant semi-algebraic sets; finite G) I SemiAlg •→• (category of morphisms) I SemiAlg I (for suitable other diagram shapes I ) background Schanuel’s presentation semi-algebraic sets

We have two functors on SemiRing:

groth : SemiRing → Ring

sch : SemiRing → JSLSemiRing with natural semiring morphisms

S → groth(S)

S → sch(S) When S is the Grothendieck semiring of a distributive category, think of S → groth(S) as the ‘universal Euler characteristic’ eu (the universal example of an abelian group valued invariant). Think of S → sch(S) as the ‘Schanuel dimension’ dim (the universal example of an idempotent semigroup valued invariant). background Schanuel’s presentation semi-algebraic sets Schanuel’s phenomenon

Schanuel’s phenomenon: For distributive categories C, the natural homomorphism

SK(C) −−−−−→dim ×eu sch SK(C) × groth SK(C)

tends to be injective.

Problem Give an example of a distributive category C whose Grothendieck semiring is finitely presentable that fails Schanuel’s phenomenon.

Schanuel hints there ought to be such.