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Schanuel Functors and the Grothendieck (Semi)Ring of Some Theories

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Schanuel Functors and the Grothendieck (Semi)Ring of Some Theories background semi-algebraic sets Schanuel functors 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 semiring of the category of semi-algebraically definable sets and functions over R. 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) 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 isomorphism 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 coproducts such that the canonical maps ? ! X × ? X × Y t X × Z ! X × (Y t Z) are isomorphisms (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 8 x y := x + y (x; y 6= −∞) > ~ > <x ~ −∞ := −∞ x y := maxfx; yg (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 := maxfx; yg 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 := f0; 1; q; q2;:::; qn;::: g x ~ y := x · y x y := maxfx; yg background distributive categories semi-algebraic sets semirings Grothendieck's functor 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 2 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 = f0g 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 f0g 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 = f0g background distributive categories semi-algebraic sets semirings examples (1) groth Nh·; +i = Zh·; +i (2) groth N−∞h+; maxi = f0g 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 2 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 2 Bn and V 2 Bm then U × V 2 Bn+m n m I if U 2 Bn then pr(U) 2 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 2 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 (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; (x; y) is a morphism from hx; α(x)i to hy; β(y)i if T proves that (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 coproduct 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.
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