Group completion in the K-theory and Grothendieck-Witt theory of proto-exact categories
aa r X i v : . [ m a t h . K T ] S e p GROUP COMPLETION IN THE K -THEORY ANDGROTHENDIECK–WITT THEORY OF PROTO-EXACTCATEGORIES JENS NIKLAS EBERHARDT, OLIVER LORSCHEID, AND MATTHEW B. YOUNG
Abstract.
We study the algebraic K -theory and Grothendieck–Witt theory ofproto-exact categories, with a particular focus on classes of examples of F -linearnature. Our main results are analogues of theorems of Quillen and Schlichting,relating the K -theory or Grothendieck–Witt theories of proto-exact categoriesdefined using the (hermitian) Q -construction and group completion. Contents
Introduction 11. Proto-exact categories with duality 32. Algebraic K -theory of uniquely split proto-exact categories 133. Grothendieck–Witt theory of uniquely split proto-exact categories 16References 26 Introduction
The algebraic K -theory of an exact category can be defined in essentially twodifferent ways. The first way, which includes Quillen’s Q -construction [14] andWaldhausen’s S • -construction [21], uses the exact structure of the category whilethe second way, which includes group completion and the +-construction [10], usesonly the underlying additive structure of the category. Each approach has its ownstrengths, both computational and theoretical. A celebrated theorem of Quillen[10], henceforth referred to as the Group Completion Theorem, asserts that the K -theories defined by the Q - or S • -constructions and by group completion agreefor split exact categories, that is, for those exact categories in which all short exactsequences are split. A closely related result, Quillen’s ‘ Q = +’ Theorem, asserts theanalogous statement for the +-construction under additional hypotheses on the cat-egory [10]. Quillen’s Group Completion Theorem was generalized to Grothendieck–Witt theory, also called hermitian K -theory, in Schlichting’s seminal work [15], [17],[18].In this paper, we investigate analogues of Quillen’s Group Completion Theoremin non-additive contexts. To do so, in place of exact categories we study proto-exact categories with an exact direct sum . This is a minimal set-up in which the GroupCompletion Theorem can be formulated. A proto-exact category, as introducedby Dyckerhoff and Kapranov [7], is a category A together with two distinguished Date : September 29, 2020.2010
Mathematics Subject Classification.
Primary: 19D10; Secondary 19G38.
Key words and phrases.
Algebraic K -theory. Grothendieck–Witt theory. Proto-exact categories. classes of morphisms, called inflations and deflations, satisfying axioms which allowfor a direct adaptation of the Q -construction and S • -construction. An exact directsum is a symmetric monoidal structure ⊕ on A which is compatible with the proto-exact structure but fulfills only a subset of the axioms of a categorical product orcoproduct.We focus our attention on uniquely split proto-exact categories, that is, proto-exact categories with exact direct sum in which each short exact sequence splitsin a unique way. Our main results are comparison theorems for K -theory andGrothendieck–Witt spaces of such categories defined using the Q -construction andgroup completion. For K -theory, our result is a direct analogue of Quillen’s GroupCompletion Theorem. The situation is more subtle for Grothendieck–Witt theoryand the results differ substantially from the exact case.The main motivation for studying uniquely split proto-exact categories is theircentral role in F -geometry [5], [4], [2]. A characteristic feature of F -geometry isthe lack of additivity in its natural constructions. For example, let A - Mod be thecategory of modules over a commutative pointed monoid A or, in geometric terms,the category of quasi-coherent sheaves over the affine monoid scheme Spec( A ). Thecategory A - Mod does not have an additive structure and so, in particular, has no ex-act structure. However, A - Mod can be given the structure of a proto-exact categorywith exact direct sum given by disjoint union. This notion is closely related to otherpartially additive structures, such as quasi-exact [5] or belian categories [6], but hasthe distinct advantage of being self-dual and so well-adapted to Grothendieck–Witttheory.A second characteristic feature of F -geometry is the distinguished role played bygeneralized monomial matrices, that is, matrices with at most one non-zero entryin each row and column. In our categorical context, this is reflected in the fact thatexact sequences in many proto-exact categories of interest not only split, but do soin a unique way. For example, the category A - proj of finitely generated projective A -modules is uniquely split proto-exact, while A - Mod is in general non-split.After establishing required categorical background in Section 1, we turn in Section2 to K -theory. Let A be a proto-exact category with exact direct sum ⊕ . Denoteby K ( A ) and K ⊕ ( A ) the K -theory spaces of A defined via the Q -construction andgroup completion, respectively. For group completion, we regard A as a symmetricmonoidal category by forgetting its proto-exact structure. The homotopy groups ofthese spaces are the corresponding algebraic K -theory groups K i ( A ) and K ⊕ i ( A ).In the setting of split exact categories, Quillen’s Group Completion Theorem [10]states that there is a homotopy equivalence K ( A ) ≃ K ⊕ ( A ). The first main result ofthis paper is a Group Completion Theorem for uniquely split proto-exact categories. Theorem A (Theorem 2.2) . Let A be a uniquely split proto-exact category. Thenthere is a homotopy equivalence K ( A ) ≃ K ⊕ ( A ) . The proof of Theorem 2.2 is a modification of the original argument [10], simpli-fied in view of the uniquely split assumption. Many of the techniques of the proofare used in the subsequent considerations for Grothendieck–Witt theory.In Section 3 we turn to Grothendieck–Witt theory, where the situation is moredelicate. Suppose that A has a compatible duality structure, that is, an exactfunctor P : A op → A together with coherence data exhibiting it as an involution.The Grothendieck–Witt space GW ( A ) is the homotopy fibre of the forgetful functor -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 3 BQ h ( A ) → BQ ( A ) over the zero object 0, where Q h ( A ) is the proto-exact hermitian Q -construction of A [1], [20]. Again, by forgetting the proto-exact structure, one canalso form a direct sum Grothendieck–Witt space GW ⊕ ( A ) using group completion.Under the assumption that A is split exact, the Group Completion Theorem forGrothendieck–Witt theory was proved only recently, giving a homotopy equivalencebetween GW ( A ) and GW ⊕ ( A ) [15], [17], [18]. The proof is considerably moreinvolved than the corresponding result for K -theory.In the setting of Grothendieck–Witt theory of uniquely split proto-exact cate-gories, we find that the Group Completion Theorem fails even in the basic case of F -vector spaces (cf. Example 3.15). However, it does hold for hyperbolic Grothendieck–Witt theory of uniquely split proto-exact categories. To state this, let Q H ( A ) ⊂ Q h ( A ) be the full subcategory on hyperbolic symmetric forms and write GW H ( A )for the homotopy fibre of BQ H ( A ) → BQ ( A ) over 0. Let also GW ⊕ H ( A ) be thespace obtained by applying group completion to the monoidal groupoid of hyper-bolic symmetric forms in A . Theorem B (Theorem 3.2) . Let A be a uniquely split proto-exact category withduality. Then there is a weak homotopy equivalence GW H ( A ) ≃ GW ⊕ H ( A ) . We use Theorem 3.2 to compute the higher Grothendieck–Witt groups of A interms of the symmetric monoidal groupoid of hyperbolic forms. Namely, we provethat there are abelian group isomorphisms GW i ( A ) ≃ GW ⊕ H,i ( A ) , i ≥ , cf. Corollary 3.9. In concrete examples, GW ⊕ H ( A ) and GW ⊕ ( A ) can often be de-scribed using the +-construction, thereby allowing for computations. Such examplesare discussed in the companion paper [8].In general, the description of the homotopy type of the full space GW ( A ) is morecomplicated than that of GW H ( A ) and GW ⊕ ( A ). Under the additional assumptionthat A is combinatorial, a hypothesis that is typically satisfied in F -linear contexts,we are able to describe GW ( A ). Here combinatorial means that any subobject U of an exact direct sum X ⊕ Y splits into objects U ∩ X ⊂ X and U ∩ Y ⊂ Y . Theorem C (Theorem 3.11) . Let A be a uniquely split combinatorial noetherianproto-exact category with duality. Then there is a weak homotopy equivalence GW ( A ) ≃ G w ∈ W ( A ) BG S w × GW H ( A ) , where S w is an isotropically simple representative of the Witt class w ∈ W ( A ) and G S w is the self-isometry group of S w . See Example 3.15 for a description of the spaces GW H ( Vect F ), GW ⊕ ( Vect F ) and GW ( Vect F ). Acknowledgements.
The authors thank Marco Schlichting for helpful correspon-dence. All three authors thank the Max Planck Institute for Mathematics in Bonnfor its hospitality and financial support.1.
Proto-exact categories with duality
In this section we establish the categorical background required for the remainderof the paper.
J. N. EBERHARDT, O. LORSCHEID, AND M. B. YOUNG
Proto-exact categories.
Proto-exact categories, introduced by Dyckerhoffand Kapranov [7, § § A with a zero object 0 together with two dis-tinguished classes of morphisms, I and D , called inflations (or admissible monomor-phisms) and deflations (or admissible epimorphisms) and denoted and ։ , re-spectively, such that the following axioms hold:(i) Any morphism 0 → U is in I and any morphism U → D .(ii) The class I is closed under composition and contains all isomorphisms, andsimilarly for D .(iii) A commutative square of the form U VW X (1)is cartesian if and only if it is cocartesian.(iv) A diagram of the form W X և V can be completed to a bicartesian squareof the form (1).(v) A diagram of the form W և U V can be completed to a bicartesian squareof the form (1).A bicartesian square (1) with W = 0 is called a conflation and, for ease of nota-tion, is denoted U V ։ X . Being kernels, inflations are necessarily monomor-phisms. Similarly, deflations are epimorphisms.If A is a proto-exact category, then the opposite category A op has a natural proto-exact structure in which the inflations (resp. deflations) are the deflations (resp.inflations) in A . The cartesian product of proto-exact categories has a naturalproto-exact structure.A functor between proto-exact categories is called proto-exact if it preserves zeroobjects and bicartesian squares of the form (1). In particular, a proto-exact functorsends conflations to conflations. Definition.
An exact direct sum on a proto-exact category A is a symmetric monoi-dal structure ⊕ on A subject to the following axioms:(DS1) The monoidal unit of A is .(DS2) The bifunctor ⊕ : A × A → A is proto-exact.Given objects
U, V ∈ A , set i U : U id U ⊕ V −−−−−−→ U ⊕ V and π U : U ⊕ V id U ⊕ V ։ −−−−−−→ U ,where we have used (DS1) to identify U ⊕ ≃ U and ⊕ V ≃ V . Axiom (DS2)implies that i U is an inflation and π U is a deflation.(DS3) The maps Hom A ( U ⊕ V, W ) → Hom A ( U, W ) × Hom A ( V, W ) , f ( f ◦ i U , f ◦ i V ) and Hom A ( W, U ⊕ V ) → Hom A ( W, U ) × Hom A ( W, V ) , f ( π U ◦ f, π V ◦ f ) -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 5 are injections for all U, V, W ∈ A .(DS4) Let U i X π V be a conflation. For each section s of π , there exists aunique isomorphism φ which makes the following diagram commute: XU U ⊕ V V. i U i φ i V s For each retraction r of i , there exists a unique isomorphism ψ which makes thefollowing diagram commute: XU U ⊕ V V. r ππ V π U ψ Remarks 1.1. (i) Motivated by the theory of Hall algebras, the notion of a proto-exact category with exact direct sum was introduced in [24, § f : L i X i → L j Y j isthe matrix ( π Y j ◦ f ◦ i X i ) i,j and this matrix uniquely determines f. However, notevery matrix of morphisms in A arises from a morphism in A . To construct a mor-phism between direct sums, one can use the symmetric monoidal structure, whichallows to construct morphisms from generalized monomial matrices whose entriesare morphisms in A , or axiom (DS4).(iii) Closely related to proto-exact categories with exact direct sum are Deitmar’squasi-exact categories [5, § ⊕ is neither required to be a product nor a coproduct.For this reason, the opposite of a quasi-exact category need not be quasi-exact, andsimilarly for belian categories, whereas if A is proto-exact with exact direct sum ⊕ ,then A op is proto-exact with exact direct sum ⊕ op . Axiom (DS4) can be seen as aweak replacement of the universal property of a product and coproduct.A functor between proto-exact categories with exact direct sum is called exactif it is proto-exact and ⊕ -monoidal. For ease of notation, we do not introducenotation for the ⊕ -monoidal data of an exact functor.In the remainder of this section, we prove some basic results about proto-exactcategories with exact direct sum.Let A be a proto-exact category with exact direct sum. A commutative diagram U X VU U ⊕ V V i πi U φ π V J. N. EBERHARDT, O. LORSCHEID, AND M. B. YOUNG of conflations with φ an isomorphism is called a splitting of U i X π V . Lemma 1.2.
Let A be a proto-exact category with exact direct sum. Then there isa bijection between the set of splittings of a conflation U i X π V , the set ofsections of π and the set of retractions of i .Proof. Given a splitting φ , the map φ ◦ i V is a section of π , as follows from thedefinitions of π V and i V . Given a section s of π , let φ : U ⊕ V → X be theisomorphism whose existence is guaranteed by axiom (DS4). Since π ◦ φ ◦ i U = π ◦ i = 0 and π ◦ φ ◦ i V = π ◦ s = id V , axiom (DS3) yields π ◦ φ = π V . In particular, φ is a splitting. That these constructions are mutually inverse follows from theuniqueness in axiom (DS4).The statements involving retractions can be proven similarly. (cid:3) Lemma 1.3.
Let φ i : U i → V i , i = 1 , , be morphisms in a proto-exact categorywith exact direct sum.(i) The square U U ⊕ U V V ⊕ V i U φ φ ⊕ φ i V commutes, as does the analogous square involving the deflations π ( − ) .(ii) The morphism φ ⊕ φ is an isomorphism if and only if φ and φ are isomor-phisms.Proof. The square of the first statement is the outside of the diagram U U ⊕ U V V ⊕ U V V ⊕ V . i U φ φ ⊕ id U i V id V ⊕ φ i V The top and bottom squares commute by the definitions of the inflations i ( − ) .For the second statement, let φ ⊕ φ be an isomorphism with inverse f , so thatid U ⊕ U = f ◦ ( φ ⊕ φ ) and id V ⊕ V = ( φ ⊕ φ ) ◦ f . These equations, together withthe first part of the lemma, give (using that π U i ◦ i U i = id U i )id U i = π U i ◦ f ◦ ( φ ⊕ φ ) ◦ i U i = π U i ◦ f ◦ i V i ◦ φ i and id V i = π V i ◦ ( φ ⊕ φ ) ◦ f ◦ i V i = φ i ◦ π U i ◦ f ◦ i V i . This shows that φ i is an isomorphism with inverse π U i ◦ f ◦ i V i . The other directionis clear. (cid:3) -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 7 Lemma 1.4.
Let A be a proto-exact category with exact direct sum. For any infla-tion j : U V , the diagram U ⊕ W V ⊕ WU V π U j ⊕ id W π V j is bicartesian. In particular, there is an isomorphism U ⊕ W ≃ U × V ( V ⊕ W ) .Proof. The diagram in question commutes by Lemma 1.3(i). By the axioms of aproto-exact category, it suffices to prove that the diagram is cocartesian. Considera commutative diagram U ⊕ W V ⊕ WU V T. π U j ⊕ id W π V rlj Since r ◦ ( j ⊕ id W ) = l ◦ π U , we see that 0 = r ◦ ( j ⊕ id W ) ◦ i W = r ◦ i W . Hence,axiom (DS3) implies that r is determined by r ◦ i V through r = r ◦ i V ◦ π V . Weclaim that u : V r ◦ i V −−→ T exhibits the universal property of a cocartesian diagram.We have u ◦ π V = r ◦ i V ◦ π V = r and u ◦ j ◦ π U = u ◦ π V ◦ ( j ⊕ id W )= r ◦ i V ◦ π V ◦ ( j ⊕ id W )= r ◦ ( j ⊕ id W )= l ◦ π U . Since π U is an epimorphism, we conclude that u ◦ j = l . (cid:3) Definition.
A proto-exact category with exact direct sum is called split (resp. uniquelysplit) if every conflation admits a splitting (resp. unique splitting).
A non-zero exact category A is never uniquely split. Indeed, for a non-zero object U ∈ A , the set of splittings of the split conflation U U ⊕ ։ U is a torsor for theadditive group End A ( U ). The uniquely split property is therefore only of interestfor non-exact proto-exact categories. Lemma 1.5.
Let A be a uniquely split proto-exact category. Given a diagram U E V U E V , i f h π gi π in A in which both rows are conflations, there exists a unique morphism h : E → E which makes the diagram commute. J. N. EBERHARDT, O. LORSCHEID, AND M. B. YOUNG
Proof.
By the uniquely split assumption, we can uniquely extend the given diagramto a commutative diagram U U ⊕ V V U E V U E V U U ⊕ V V i U π V φ i f π gi π φ i U π V for some isomorphisms φ and φ . The reduces the problem to the case U U ⊕ V V U U ⊕ V V , i U f π V gi U π V in which we can take h = f ⊕ g . That h is the unique admissible choice followsfrom axiom (DS3). (cid:3) In particular, it follows from Lemma 1.5 that for a diagram of conflations of theform
U E VU E V, i π i π there exists a unique morphism h : E → E making the diagram commute andthat, moreover, that h is an isomorphism.Motivated by [9], we make the following definition. See also [24, § Definition.
A proto-exact category with exact direct sum is called combinatorial if,for each inflation i : U X ⊕ X , there exist inflations i k : U k X k , k = 1 , ,and an isomorphism f : U → U ⊕ U such that i = ( i ⊕ i ) ◦ f and, dually, foreach deflation π : X ⊕ X ։ U , there exist deflations π k : X k U k , k = 1 , , andan isomorphism g : U ⊕ U → U such that π = g ◦ ( π ⊕ π ) . Given an inflation i : U X ⊕ X , we sometimes write U ∩ X k for the object U k , k = 1 , Lemma 1.6.
Let A be uniquely split combinatorial proto-exact category. If φ : U X ⊕ Y is an inflation such that π X ◦ φ is an inflation, then π Y ◦ φ = 0 . Dually, if φ : X ⊕ Y ։ U is a deflation such that φ ◦ i X is a deflation, then φ ◦ i Y = 0 . -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 9 Proof.
The combinatorial assumption implies that there is a decomposition U ≃ U X ⊕ U Y under which φ becomes a morphism φ X ⊕ φ Y : U X ⊕ U Y X ⊕ Y. Then we have π X ◦ φ = φ X ◦ π U X . Since π X ◦ φ is an inflation, the kernel U Y of π U X : U X ⊕ U Y ։ U X is trivial. It follows that π Y ◦ φ = φ Y ◦ π U Y = 0. The secondstatement can be proved in the same way. (cid:3) Example 1.7.
In our companion paper [8], we study proto-exact categories oc-curring in F -geometry. These categories come typically with an exact direct sumand are uniquely split and combinatorial. We discuss this in the simplest case ofa F -linear category, which is the category Vect F of “ F -vector spaces”. Its ob-jects are pointed sets, which come with a tautological action of the pointed monoid F = { , } . Its morphisms are base point preserving maps that are injective outsidethe fibre over the base point.The direct sum N ⊕ M = N ∨ M of two pointed sets N and M is the wedge sum,that is, their disjoint union modulo the identification the base points. It comes withcanonical inclusions and projections, the latter contracting the opposite summandto the base point. It is an easy exercise to verify that every conflation in Vect F isof the form N i N M ⊕ N π M M, so that that Vect F is uniquely split. Similarly, it is easy to see that Vect F iscombinatorial.1.2. Proto-exact categories with duality.
For a detailed introduction to cat-egories with duality, the reader is referred to [16, § A , P, Θ) consisting of a category A , a functor P : A op → A and a natural isomorphism Θ : id A ⇒ P ◦ P op which satisfies P (Θ U ) ◦ Θ P ( U ) = id P ( U ) , U ∈ A . We often omit the pair ( P, Θ) from the notation if it will not cause confusion. If A has a proto-exact structure and P is proto-exact, then we call this a proto-exactcategory with duality.A (nondegenerate) symmetric form in ( A , P, Θ) is a pair (
M, ψ M ) consisting of anobject M ∈ A and an isomorphism ψ M : M → P ( M ) which satisfies P ( ψ M ) ◦ Θ M = ψ M . We often write M or ψ M for ( M, ψ M ). An isometry φ : ( M, ψ M ) → ( N, ψ N )is an isomorphism φ : M → N which satisfies ψ M = P ( φ ) ◦ ψ N ◦ φ . The groupoidof symmetric forms and their isometries is denoted S h and called the hermitiangroupoid of A .Let ( M, ψ M ) be a symmetric form in a proto-exact category with duality. Aninflation i : U M is called isotropic if P ( i ) ◦ ψ M ◦ i is zero and the inducedmonomorphism U → U ⊥ := ker( P ( i ) ◦ ψ M ) is an inflation. In this case, M//U := U ⊥ /U , the isotropic reduction of M by U , inherits a symmetric morphism ψ M//U : M//U → P ( M//U ).We say that ( A , P, Θ) satisfies the Reduction Assumption (as introduced in [23, § M, ψ M ) and every isotropic inflation i : U M ,the symmetric morphism ψ M//U : M//U → P ( M//U ) is an isomorphism. Exactcategories satisfy the Reduction Assumption [16, Lemma 2.6], as do many proto-exact categories [24], [8]. A symmetric form (
M, ψ M ) is called metabolic if it has a Lagrangian, that is, an isotropic subobject U M with U = U ⊥ , and is calledisotropically simple if it has no non-zero isotropic subobjects.In the following, we assume that A has an exact direct sum, that P is exactand that Θ is a ⊕ -monoidal natural isomorphism. In this case, we have P ( i U ) = π P ( U ) and P ( π U ) = i P ( U ) for each U ∈ A . With these assumptions, ⊕ defines anorthogonal direct sum of symmetric forms by( M, ψ M ) ⊕ ( N, ψ N ) = ( M ⊕ N, ψ M ⊕ ψ N ) . This gives S h the structure of a symmetric monoidal groupoid. Moreover, ⊕ allowsto define hyperbolic symmetric forms. Namely, given an object U ∈ A , the pair (cid:16) H ( U ) = U ⊕ P ( U ) , ψ H ( U ) = (cid:16) P ( U ) Θ U (cid:17)(cid:17) is a symmetric form in A , called the hyperbolic form on U . The assignment U H ( U ) extends to a functor H : S → S h , where S is the maximal groupoid of A . Asymmetric form is called hyperbolic if it is isometric to one of the form ψ H ( U ) .We now prove an analogue of the fact that split metabolics in an exact categorywith duality in which “2 is invertible” are hyperbolic [12, Lemma 2.5(b)]. Lemma 1.8.
A metabolic form in a uniquely split proto-exact category with dualityis hyperbolic.Proof.
Let i : U ( M, ψ M ) be a Lagrangian. We can extend i to a commutativediagram U M P ( U ) P ( U ) P ( M ) P ( U ) . i Θ U P ( i ) ◦ ψ M ψ M ψ M ◦ i ◦ Θ − U P ( i ) Since A is (uniquely) split, this diagram can be extended to the commutative dia-gram U U ⊕ P ( U ) P ( U ) U N P ( U ) P ( U ) P ( M ) P ( U ) P ( U ) P ( U ⊕ P ( U )) P ( U ) i U π U φi Θ U P ( i ) ◦ ψ M ψ M ψ M ◦ i ◦ Θ − U P ( i ) P ( φ ) P ( π U ) P ( i U ) for some isomorphism φ . Then P ( φ ) ◦ ψ M ◦ φ and ψ H ( U ) are endomorphisms of thesplit conflation on U and P ( U ) and so, by Lemma 1.5, are equal. Hence, φ is anisometry ψ H ( U ) ∼ −→ ψ M . (cid:3) Proposition 1.9.
Let ( N, ψ N ) be a symmetric form in a uniquely split proto-exactcategory with duality and i : M N an inflation such that ψ M := P ( i ) ◦ ψ N ◦ i is a symmetric form on M . Then there exists a symmetric form ( M ′ , ψ M ′ ) and anisometry ( N, ψ N ) ≃ ( M, ψ M ) ⊕ ( M ′ , ψ M ′ ) which identifies i with i M : M M ⊕ M ′ . -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 11 Proof.
Set π := ψ − M ◦ P ( i ) ◦ ψ N : N ։ M and define M ′ ∈ A by the conflation M ′ j N π M . Set ψ M ′ = P ( j ) ◦ ψ N ◦ j .Fix a splitting M ′ N MM ′ M ′ ⊕ M M j πi M ′ π M φ and define a symmetric form on M ′ ⊕ M by ψ = P ( φ ) ◦ ψ N ◦ φ . With thesedefinitions, we have P ( i M ′ ) ◦ ψ ◦ i M ′ = ψ M ′ and P ( i M ) ◦ ψ ◦ i M = P ( φ ◦ i M ) ◦ ψ N ◦ ( φ ◦ i M ) = ψ M . To see the last equality, note that since φ ◦ i M and i are sections of π , uniquenessof splittings and Lemma 1.2 combine to give φ ◦ i M = i . We also have P ( i M ) ◦ ψ ◦ i M ′ = P ( φ ◦ i M ) ◦ ψ N ◦ j = P ( i ) ◦ ψ N ◦ j = 0and, dually, P ( i M ′ ) ◦ ψ ◦ i M = 0. Using axiom (DS3), we conclude that ψ = ψ M ⊕ ψ M ′ .Finally, ψ M ′ is symmetric by construction and an isomorphism by Lemma 1.3.Hence, ( M ′ , ψ M ′ ) is a symmetric form. (cid:3) Proposition 1.10.
Let ( N, ψ N ) be a symmetric form in a uniquely split proto-exactcategory with duality. If i : U ( N, ψ N ) is isotropic, then there exists an isometry φ : N → H ( U ) ⊕ ( N//U ) which identifies U U ⊥ N with U i U U ⊕ ( N//U ) i U ⊕ ( N//U ) H ( U ) ⊕ ( N//U ) . Proof.
Set M = N//U . Complete i to the following diagram, all of whose rectanglesare bicartesian: U U ⊥ N M P ( U ⊥ )0 P ( U ) . k jπ P ( j ) ◦ ψ N P ( π ) ◦ ψ M P ( k ) Here i = j ◦ k . Let f : U ⊥ ∼ −→ U ⊕ M be the unique splitting of U U ⊥ ։ M and set j ′ = j ◦ f − . Let φ : N ∼ −→ U ⊕ P ( U ) ⊕ M be the unique splitting of U ⊕ M j ′ N ։ P ( U ) and set ψ ′ = P ( φ − ) ◦ ψ N ◦ φ − . Then the previous diagram becomes U U ⊕ M U ⊕ P ( U ) ⊕ M M P ( U ⊕ M )0 P ( U ) . i U i U ⊕ M π M P ( i U ⊕ M ) ◦ ψ ′ P ( π M ) ◦ ψ M P ( i U ) In particular, we have P ( π M ) ◦ ψ M ◦ π M = P ( i U ⊕ M ) ◦ ψ ′ ◦ i U ⊕ M . Pre- and post-composing this equation with i M : M U ⊕ M and P ( i M ), respec-tively, gives P ( i M ) ◦ P ( π M ) ◦ ψ M ◦ π M ◦ i M = P ( i M ) ◦ P ( i U ⊕ M ) ◦ ψ ′ ◦ i U ⊕ M ◦ i M , which can be rewritten as ψ M = P ( i M U ⊕ P ( U ) ⊕ M ) ◦ ψ ′ ◦ i M U ⊕ P ( U ) ⊕ M . Proposition 1.9 therefore implies that there exists an isometry( U ⊕ P ( U ) ⊕ M, ψ ) ≃ ( U ⊕ P ( U ) , ψ U ⊕ P ( U ) ) ⊕ ( M, ψ M )under which U U ⊕ P ( U ) ⊕ M factors through the standard Lagrangian i U : U ( U ⊕ P ( U ) , ψ U ⊕ P ( U ) ). We conclude using Lemma 1.8 that ψ U ⊕ P ( U ) ≃ ψ H ( U ) . (cid:3) A proto-exact category is called noetherian if any ascending chain of inflationsstabilizes after finitely many steps.
Proposition 1.11.
Let ( M, ψ M ) be a symmetric form in a uniquely split noetherianproto-exact category A with duality.(i) There exists an object U ∈ A , an isotropically simple symmetric form ( N, ψ N ) ∈S h and an isometry M ≃ H ( U ) ⊕ N .(ii) If, moreover, A is combinatorial, then the decomposition from part (i) is uniqueup to isometry in either summand.Proof. If M is isotropically simple, then take U = 0 and we are done. Otherwise, let U M be a non-zero isotropic. Then M ≃ H ( U ) ⊕ M by Proposition 1.10. If M is isotropically simple, then we are done. Otherwise, choose a non-zero isotropic˜ U M , so that M ≃ H ( U ) ⊕ M with U = U ⊕ ˜ U . Continuing in this way,we obtain an ascending chain of inflations U U · · · of M , which is finite bythe noetherian hypothesis. This proves the first part.For the second part, let φ : H ( U ) ⊕ N → H ( U ′ ) ⊕ N ′ be an isometry with N , N ′ isotropically simple. Then φ ◦ i U : U H ( U ′ ) ⊕ N ′ is isotropic and hence so tooare the subobjects φ ( U ) ∩ H ( U ′ ) and φ ( U ) ∩ N ′ . Since N ′ is isotropically simple, -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 13 φ ( U ) ∩ N ′ = 0, whence φ ◦ i U factors through H ( U ′ ). Similarly, φ ◦ i P ( U ) factorsthrough H ( U ′ ). We therefore have a commutative diagram H ( U ) H ( U ) ⊕ N NH ( U ′ ) H ( U ′ ) ⊕ N ′ N ′ . i H ( U ) φ L π N φ φ R i H ( U ′ ) π N ′ (2)By applying the same argument to φ − and patching the resulting diagram withdiagram (2) (similar to the proof of Lemma 1.8), we conclude that φ L and φ R areisomorphisms. Since φ is symmetric, so too are φ L and φ R . (cid:3) Example 1.12.
An exact duality for the category
Vect F from Example 1.7 is thefunctor P : Vect op F → Vect F that is the identity on objects and maps a morphism f : M → N to its adjoint f ad : N → M , defined by f ad ( n ) = m if n = f ( m ) and f ad ( n ) = 0 if n / ∈ im( f ), where 0 is the base point of M . The natural isomorphismΘ is the identity between id Vect F and P ◦ P op = id Vect F .2. Algebraic K -theory of uniquely split proto-exact categories We study K -theory spaces of proto-exact and symmetric monoidal categories.Our main result, Theorem 2.2, is a Group Completion Theorem for uniquely splitproto-exact categories. The techniques and results of this section are used to studyGrothendieck–Witt theory in Section 3.2.1. The proto-exact Q -construction. Let A be a proto-exact category. Sim-ilarly to the exact case [14, § Q -construction of A is the category Q ( A ) defined as follows. An object of Q ( A ) is simply an object of A . A morphism U → V in Q ( A ) is an equivalence class of diagrams U և E V in A . Two suchdiagrams U և E V and U և E ′ V are equivalent if there exists an isomor-phism E → E ′ in A which makes the obvious diagrams commute. Composition ofmorphisms in Q ( A ) is defined via pullback and is well-defined by the axioms of aproto-exact category. The K -theory space of A is K ( A ) = Ω BQ ( A ) , the based loop space of the classifying space of Q ( A ), where 0 ∈ Q ( A ) determinesthe basepoint of BQ ( A ). The algebraic K -theory groups of A are the homotopygroups K i ( A ) = π i K ( A ) , i ≥ . We remark that an exact direct sum on A induces a symmetric monoidal structureon Q ( A ). Example.
Let A be a proto-exact category. Then K ( A ) is isomorphic to the freegroup on isomorphism classes of objects of A modulo the relation [ V ] = [ U ][ W ]whenever U V ։ W . In general, the group K ( A ) need not be abelian [11,Example 3.6.3]. If, however, A admits an exact direct sum, then by using the splitconflation on U and W we see that [ U ][ W ] = [ W ][ U ]. In this case, K ( A ) is abelianand is described in the familiar way. ⊳ Direct sum K -theory. Let ( A , ⊕ ) be a symmetric monoidal category. Notethat proto-exact categories with exact direct sum and exact categories define sym-metric monoidal categories by forgetting the (proto-)exact structure. The maximalgroupoid S of A is symmetric monoidal. The direct sum K -theory space of A isthe group completion of B S : K ⊕ ( A ) = B ( S − S ) . See, for example, [10, Page 222]. We refer the reader to [22, § IV.4] for our conven-tions on group completion.
Example 2.1.
We calculate the K -theory of certain F -linear categories in ourcompanion paper [8]. In the simplest case, that of Vect F (see Example 1.7), wefind K ⊕ ( Vect F ) ≃ Z × ( B Σ ∞ ) + , as a special case of [8, Theorem 2.5].2.3. A Group Completion Theorem for K -theory of uniquely split proto-exact categories. Quillen’s Group Completion Theorem [10] states that for anysplit exact category A , there is a homotopy equivalence K ( A ) ≃ K ⊕ ( A ). In thissection, we prove a proto-exact version of this result. Theorem 2.2.
Let A be a uniquely split proto-exact category. Then there is ahomotopy equivalence K ( A ) ≃ K ⊕ ( A ) . It is claimed (without proof) in [3, §
4] that the proof from [10] for split exactcategories can be modified to apply to (not necessarily uniquely) split proto-exactcategories. Instead of following this line of thought, we give a simplified proof in theuniquely split case, which is the setting of interest for this paper. These argumentsare used in Section 3.We begin with some preparatory material. Write Q for the category Q ( A ) andlet E = E ( A ) be the category of conflations in A . Objects of E are conflations.Morphisms in E (from primed to unprimed conflations) are equivalence classes ofcommutative diagrams A ′ B ′ C ′ A B ′ C A B C in A whose rows are conflations. The equivalence relation on morphisms is generatedby automorphisms of C . Projection to the right column defines a fibered functor g : E → Q . Given C ∈ Q , let E C := g − ( C ) be the fibre category. The symmetricmonoidal groupoid S acts on E by A ′ · ( A i B π C ) = ( A ′ ⊕ A id A ′ ⊕ i A ′ ⊕ B π C ) . Since the functor ⊕ is proto-exact, the right hand side is indeed a conflation. The S -action restricts to the fibres of g and, when Q is equipped with the trivial S -action, the base change maps of g are S -equivariant. The functor g is thereforecartesian; see [10, Pages 222 and 226]. -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 15 The uniquely split assumption on A allows for the following explicit descriptionof the fibers of g . We note that there is no such description for exact categories. Lemma 2.3.
The functor F C : S → E C , given on objects and morphisms by A ( A i A C ⊕ A π C C ) and ( φ : A ′ ∼ −→ A ) A ′ C ⊕ A ′ CA C ⊕ A ′ CA C ⊕ A C i A ′ π C φ − i A ′ ◦ φ − id C ⊕ φ π C i A π C respectively, is an S -equivariant equivalence.Proof. Since A is (uniquely) split, F C is essentially surjective. That F C is faithfulfollows from the definitions. To see that F C is full, note that a morphism f : F C ( A ′ ) → F C ( A ) in E C is represented by a commutative diagram A ′ C ⊕ A ′ CA C ⊕ A ′ CA C ⊕ A C. i A ′ π C α i β π C i A π C Since i A ′ and i = i A ′ ◦ α are kernels of π C : C ⊕ A ։ C , the map α is an isomorphism.Post-composing the equation i A = β ◦ i A ′ ◦ α with π A gives id A = ( π A ◦ β ◦ i A ′ ) ◦ α ,from which we conclude that α − = π A ◦ β ◦ i A ′ . It follows that both β and id C ⊕ α − define morphisms of conflations A ′ C ⊕ A ′ CA ′ C ⊕ A C. i A ′ π C β id C ⊕ α − i A ◦ α − π C Applying Lemma 1.5, we conclude that β = id C ⊕ α − , so that f = F C ( α − ). Hence, F C is an equivalence.The S -equivariance of F C follows from the properties of ⊕ . (cid:3) The morphism z C = (0 և C ) in Q induces a base change functor z ∗ C : E C →E , given on objects by z ∗ C ( A B ։ C ) = ( A id A A p C = (0 և C id C C )induces a base change functor p ∗ C : E C → E , given on objects by p ∗ C ( A B ։ C ) = ( B id B B . There is also a functor p C ∗ : E → E C given on objects by p C ∗ ( A B ։
0) = (
A C ⊕ B π C C ) . Lemma 2.4. (i) The functor z ∗ C is an equivalence.(ii) There are natural isomorphisms of functors C · ≃ p ∗ C p C ∗ : E → E and C · ≃ p C ∗ p ∗ C : E C → E C , where C · denotes the action of C ∈ S .Proof. It is straightforward to verify that there is a natural isomorphism z ∗ C F C ≃ F . Since F C and F are equivalences (Lemma 2.3), so too is z ∗ C . The secondnatural isomorphism is proved similarly, using instead that p ∗ C p C ∗ F ≃ C · F and p C ∗ p ∗ C F C ≃ C · F C . (cid:3) Denote by ˜ g the composition S − E S − g −−−→ S − Q −→ Q , the second functor beingthe equivalence induced by the projection S × Q → Q . Lemma 2.5.
After passing to classifying spaces, the sequence S − E → S − E ˜ g −→ Q is a homotopy fibration.Proof. Since g is cartesian, the fibres of ˜ g are of the form S − E C and the base changemap for a morphism f : C ′ → C in Q is S − f ∗ , where f ∗ : E C → E C ′ is the basechange for g ; see [10, Page 222]. To show that all base change maps are equivalencesit suffices to consider the morphisms z C and p C ; cf . [19, Lemma 7.9]. By Lemma2.4, the functor z ∗ C is an equivalence, hence so too is S − z ∗ C . Moreover, S − p C ∗ isquasi-inverse to S − p ∗ C up to multiplication by C . Since multiplication by C is anequivalence on the S -localized categories, Quillen’s Theorem B [14, §
1] implies thatthe sequence in question is a homotopy fibration. (cid:3)
Proof of Theorem 2.2.
The space B S − E is contractible, as can be seen in the sameway as for the exact case [10, Page 228]. The homotopy fibration of Lemma 2.5therefore implies that Ω BQ is homotopy equivalent to B S − E . By Lemma 2.3, thecategories S − E and S − S are equivalent. The theorem follows. (cid:3) Grothendieck–Witt theory of uniquely split proto-exactcategories
The subject of this section is the Grothendieck–Witt spaces of proto-exact andsymmetric monoidal categories with duality. Our first result, Theorem 3.2, is ahyperbolic Group Completion Theorem for uniquely split proto-exact categorieswith duality. We then use this result to describe the connected components of theGrothendieck–Witt space defined via the hermitian Q -construction. The result isTheorem 3.11. -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 17 The proto-exact hermitian Q -construction. The Grothendieck–Witttheory of an exact category with duality can be defined using the hermitian Q -construction [1, § § § A , P, Θ) to define a category Q h ( A ). Anobject of Q h ( A ) is a symmetric form in A . A morphism ( M, ψ M ) → ( N, ψ N ) in Q h ( A ) is an equivalence class of diagrams M π և −− E j N in A such that j iscoisotropic, that is, U := ker( P ( j ) ◦ ψ N ) N is isotropic, and the diagram E NM P ( E ) jπ P ( j ) ◦ ψ N P ( π ) ◦ ψ M is bicartesian. In words, a morphism M → N in Q h ( A ) is a presentation of M asan isotropic reduction of N . Equivalence and composition of morphisms is as in Q ( A ). Denote by F : Q h ( A ) → Q ( A ) the forgetful functor.The Grothendieck–Witt space GW ( A ) is the homotopy fibre of BF : BQ h ( A ) → BQ ( A ) over 0. The Grothendieck–Witt groups of A are the homotopy groups GW i ( A ) = π i GW ( A ) , i ≥ . Despite the name, we note that, without further assumptions, GW ( A ) is in factonly a pointed set. The higher Witt groups of A are W i ( A ) = coker (cid:16) K i ( A ) π i H ∗ −−−→ GW i ( A ) (cid:17) , i ≥ H ∗ : K ( A ) = Ω BQ ( A ) → GW ( A ) is the natural map, defined up to homo-topy).Suppose now that A has an exact direct sum ⊕ . This induces symmetric monoidalstructures on S h and Q h ( A ) and F extends to a symmetric monoidal functor. Inthis case, GW ( A ) is a commutative monoid. Define the Witt monoid W ( A ) to bethe monoid π ( S h ) of isometry classes of symmetric forms modulo the submonoidof metabolic symmetric forms. Then W ( A ) is a commutative monoid. For acomparison of this definition with that of W i ( A ), i ≥
1, see the comments belowCorollary 3.14.
Example.
Suppose that A is an exact category. In this case W ( A ) is a group.Indeed, the inverse of [( M, ψ M )] ∈ W ( A ) is [( M, − ψ M )], since the diagonal map M ∆ ψ M ⊕ − ψ M is Lagrangian. Similarly, GW ( A ) is an abelian group [16,Proposition 4.11]. ⊳ Remark 3.1.
Alternatively, in Sections 2.1 and 3.1 we could have used the proto-exact Waldhausen S • -construction S • ( A ) [21, § § R • -construction R • ( A ) [13, § § K -theory and Grothendieck–Witt theory spaces of A , respectively. The resulting spaces are homotopy equivalentto those defined using the Q -constructions in a way which is compatible with themaps H ∗ and F . Direct sum Grothendieck–Witt theory.
Let ( A , ⊕ ) be a symmetricmonoidal category with duality. Then S h is a symmetric monoidal groupoid. Fol-lowing [12, § A is the group comple-tion GW ⊕ ( A ) = B ( S − h S h ) . The direct sum Grothendieck–Witt and Witt groups are GW ⊕ i ( A ) = π i GW ⊕ ( A ) , i ≥ W ⊕ i ( A ) = coker( K ⊕ i ( A ) H −→ GW ⊕ i ( A )) , i ≥ H is induced by the map K ⊕ ( A ) → GW ⊕ ( A ) de-termined by the symmetric monoidal hyperbolic functor H : S → S h . Note that GW ⊕ i ( A ) and W ⊕ i ( A ), i ≥
0, are abelian groups.3.3.
A hyperbolic Group Completion Theorem for Grothendieck–Witttheory.
In this section we initiate the study of the relation between GW ( A ) and GW ⊕ ( A ) for a uniquely split proto-exact category A . In the split exact case, thespaces GW ( A ) and GW ⊕ ( A ) are homotopy equivalent, thereby giving a hermitiananalogue of the Group Completion Theorem; see [15, Theorem 4.2], [17, TheoremA.1] when “2 is invertible” and [18, Theorem 6.6] in general. While the naiveanalogue of the Group Completion Theorem for Grothendieck–Witt theory fails inthe proto-exact setting (cf. Example 3.15), a hyperbolic analogue does indeed hold.To begin, we introduce some notation. Let S H ⊂ S h and Q H ( A ) ⊂ Q h ( A ) be thefull subcategories of hyperbolic objects. Denote by GW H ( A ) the homotopy fibre of BF : BQ H ( A ) → BQ ( A ) over 0 and set GW ⊕ H ( A ) = B ( S − H S H ). We often omit A from the notation so that, for example, Q H = Q H ( A ). Theorem 3.2.
Let A be a uniquely split proto-exact category with duality. Thenthere is a weak homotopy equivalence GW H ( A ) ≃ GW ⊕ H ( A ) . The proof of Theorem 3.2 occupies the remainder of this section.
Remark 3.3.
In [1, Theorem] a Group Completion Theorem for Grothendieck–Witt theory of exact categories in which 2 is invertible is claimed. However, theproof contains a serious error which does not appear to be fixable; see [13, Section 2,Remark]. Correct proofs were found only much later [15], [17], [18]. However, sincethese approaches rely on the additive structure of exact categories in an essentialway, they do not carry over to the proto-exact setting. Instead, it turns out that,by restricting to proto-exact categories which are uniquely split, we can adapt partsof the strategy of [1] while avoiding the critical mistake.Let τ : S H → Q H be the functor which is the identity on objects and sends anisometry M φ −→ N to M id M և −−−− M φ N . We begin by studying the (right)comma categories of τ . Fix M ∈ Q H . The groupoid S acts on the comma category M \ τ by V · ( N, ( M q և E j N )) = (cid:18) H ( V ) ⊕ N, ( M q ◦ π E և −−−− V ⊕ E i V ⊕ j H ( V ) ⊕ N ) (cid:19) . The following lemma, which relies on the uniquely split assumption and is not truein the exact setting, gives an explicit description of M \ τ . -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 19 Lemma 3.4.
The functor H M : S → M \ τ , given on objects by H M ( V ) = (cid:18) M ⊕ H ( V ) , f MV = ( M π M և −−− M ⊕ V id M ⊕ i V M ⊕ H ( V )) (cid:19) and morphisms in the obvious way, is an equivalence.Proof. Essential surjectivity of H M follows from Proposition 1.10. Faithfulness of H M follows from that of the hyperbolic functor H : S → S H . To see that H M isfull, let φ : H M ( U ) → H M ( V ) be a morphism, that is, an isometry φ : M ⊕ H ( U ) → M ⊕ H ( V ) such that φ ◦ f MU = f MV as morphisms in Q H . Since φ ◦ f MU = ( M π M և −−− M ⊕ U φ ◦ i M ⊕ U M ⊕ H ( V )) , the equality φ ◦ f MU = f MV implies the existence of an isomorphism α : M ⊕ U → M ⊕ V such that π M ⊕ V → M ◦ α = π M ⊕ U → M and (id M ⊕ i V ) ◦ α = φ ◦ i M ⊕ U . By thefirst equality, there exists a commutative diagram U M ⊕ U MV M ⊕ V M. i U ˜ α π M αi V π M Since α is an isomorphism, so too is ˜ α . Using Lemma 1.5, we conclude that α =id M ⊕ ˜ α . Using the second equality, we then verify that φ = id M ⊕ H ( ˜ α ) =: H M ( ˜ α ).The S -equivariance of H M is clear. (cid:3) Let K ∈ S . Then 0 K := (0 և K i K H ( K )) is a morphism 0 → H ( K ) in Q H .The base change functor 0 ∗ K : H ( K ) \ τ → \ τ is given on objects by0 ∗ K ( N, ( H ( K ) q և E j N )) = ( N, (0 և −−− K × H ( K ) E i K × M j N )) . Consider also the functor 0 K ∗ : 0 \ τ → H ( K ) \ τ given on objects by0 K ∗ ( N, (0 և E j N )) = ( H ( K ) ⊕ N, ( H ( K ) π H ( K ) և −−−−− H ( K ) ⊕ E id H ( K ) ⊕ j H ( K ) ⊕ N ))and on morphisms in the obvious way. Lemma 3.5.
There are natural isomorphisms of functors K · ≃ ∗ K K ∗ : 0 \ τ → \ τ and K · ≃ K ∗ ∗ K : H ( K ) \ τ → H ( K ) \ τ, where K · denotes the action of K ∈ S on Q H . Proof.
For ease of notation, set M = H ( K ). Fix A = ( N, (0 և E j N )) ∈ \ τ .We compute0 ∗ K K ∗ ( A ) = (cid:16) M ⊕ N, (0 և K × M ( M ⊕ E ) id K × M (id M ⊕ j ) M ⊕ N ) (cid:17) = (cid:16) M ⊕ N, (0 և K ⊕ E i K ⊕ j M ⊕ N ) (cid:17) = K · (cid:16) N, (0 և E j N ) (cid:17) = K · A. Lemma 1.4 was used to deduce the second equality. These identifications are clearlynatural in A and the first claimed natural isomorphism follows.In view of Lemma 3.4, to establish the second natural isomorphism it suffices toshow that K · H M ≃ K ∗ ∗ K H M . Let V ∈ S and set N = H ( V ). Then we have0 K ∗ ∗ K H M ( V ) = 0 K ∗ (cid:16) M ⊕ N, (0 և K ⊕ V i K ⊕ i V M ⊕ N ) (cid:17) = (cid:16) M ⊕ M ⊕ N, (0 և M ⊕ K ⊕ E id M ⊕ i K ⊕ i V M ⊕ M ⊕ N ) (cid:17) while K · H M ( V ) = (cid:16) M ⊕ M ⊕ N, (0 և K ⊕ M ⊕ E i K ⊕ id M ⊕ i V M ⊕ M ⊕ N ) (cid:17) . The map s V : M ⊕ M ⊕ N → M ⊕ M ⊕ N , which swaps the first two summands and isthe identity on the third, provides an isomorphism s V : K · H M ( V ) → K ∗ ∗ K H M ( V )which is evidently natural in V . (cid:3) Remark 3.6.
The mistake in the argument of [1] is related to the swap naturaltransformation s V . In [1] it is claimed that the pullback σ ∗ : ( M ⊕ M ) \ τ → ( M ⊕ M ) \ τ associated to the swap σ : M ⊕ M → M ⊕ M is an inner automorphism(which is not true in general) and hence induces the identity on homology; see [13,Remark 1.5]. Our argument, which relies in a critical manner on the uniquely splitassumption, uses the explicit description of the categories M \ τ in Lemma 3.4 anddoes not require that σ ∗ be inner.The groupoid S acts on S H and Q H via the hyperbolic functor H : S → S H .With these actions, τ : S H → Q H is S -equivariant. Consider the localized functor S − τ : S − S H → S − Q H . The natural inclusion Q H → S − Q H admits a quasi-inverse λ , given on objects and morphisms by λ ( V, M ) = M and λ ( X, X ⊕ V α −→ V ′ , H ( X ) ⊕ N β −→ N ′ ) = β ◦ ( N π X և −−− X ⊕ N i X ⊕ N H ( X ) ⊕ N ) , respectively. Write θ : S − S H → Q H for the composition λ ◦ S − τ . Lemma 3.7.
For any morphism β : M ′ → M in Q H , the base change functor β ∗ : M \ θ → M ′ \ θ induces a homotopy equivalence on classifying spaces.Proof. There is an equivalence Φ M : S − ( M \ τ ) → M \ θ ; cf . the proof of [1, Lemma3.2]. On objects, set Φ M ( U, ( N, f )) = ((
U, N ) , f ). Consider a morphism( V ; γ, δ ) : ( U, ( N, f )) → ( U ′ , ( N ′ , f ′ )) -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 21 in S − ( M \ τ ). Here V ∈ S and γ : V ⊕ U ∼ −→ U ′ and δ : H ( V ) ⊕ N ∼ −→ N ′ is anisometry which satisfies β ◦ f = f ′ . Set Φ M ( V ; γ, δ ) = ( V ; γ, δ ). It is straightforwardto verify that the following diagram commutes up to natural isomorphism: S − ( M \ τ ) S − ( M ′ \ τ ) M \ θ M ′ \ θ. Φ M S − ( β ∗ ) Φ M ′ β ∗ It therefore suffices to prove that S − ( β ∗ ) induces an equivalence on classifyingspaces. Let K ′ M ′ be a Lagrangian and set K := K ′ × M ′ E M . We then have0 K ◦ β = 0 K ′ and, by the 2-out-of-3 property for homotopy equivalences, it sufficesto consider the case β = 0 K . Lemma 3.5 implies that S − (0 K ∗ ) and S − (0 ∗ K ) arequasi-inverse up to S − ( K · ). However, S − ( K · ) is homotopic to the identity byproperties of localization. The lemma follows. (cid:3) Lemma 3.8.
The sequence B ( S − S ) B ( S − H ) −−−−−→ B ( S − S H ) Bθ −−−−−→ BQ H is a homotopy fibration.Proof. In view of Lemma 3 .
7, Quillen’s Theorem B applies to θ , allowing us toconclude that Bθ is a homotopy fibration with homotopy fibre B (0 \ θ ). Lemma 3.4and the equivalence Φ M from the proof of Lemma 3.7 compose to an equivalence S − S S − H M −−−−→ S − ( M \ τ ) Φ M −−−−→ M \ θ. Setting M = 0 completes the proof. (cid:3) Proof of Theorem 3.2.
We follow the strategy of proof from the exact setting [13,Theorem 3.5]. Consider the diagram S − E S − E × Q Q H S − E Q H Q ˜ g ∗ y ˜ gF whose square is cartesian and right column is as in Lemma 2.5. Since ˜ g is fibredand fulfills the conditions of Quillen’s Theorem B and S − E is contractible, we canapply [13, Proposition 3.4] to conclude that (after passing to classifying spaces) thesequences S − E ι −→ S − E × Q Q H ˜ g ∗ −→ Q H and S − E × Q Q H ˜ g ∗ −→ Q H F −→ Q are homotopy fibrations. Consider the diagram S − S S − S H Q H S − E S − E × Q Q H Q H , S − H S − F θJι ˜ g ∗ where F is the equivalence from Lemma 2.3 and J is defined as in [13, Theorem3.5]. The right square commutes and the left square commutes up to natural iso-morphism. By Lemma 3.8, the top row is a homotopy fibration. Since S − F is anequivalence, it follows from the five lemma that BJ is a weak homotopy equivalence.The above discussion gives a weak homotopy equivalence GW H ≃ B ( S − E ) × BQ BQ H . Moreover, there is a homotopy equivalence B ( S − S H ) ≃ B ( S − H S H ) = GW ⊕ H .Composing with BJ then gives the desired weak homotopy equivalence GW H ≃GW ⊕ H . (cid:3) The above arguments show that there is a homotopy commutative diagram B ( S − S ) B ( S − S H ) BQ H ( A ) BQ ( A )Ω BQ ( A ) GW H ( A ) BQ H ( A ) BQ ( A )Ω BQ ( A ) GW ( A ) BQ h ( A ) BQ ( A ) B ( S − H ) Bθ BFBFH ∗ BF whose rows are homotopy fibre sequences. Using Theorem 2.2 we conclude that thefirst two rows are weak homotopy equivalent. It follows that the morphism H ∗ fromequation (3) is explicitly realized by the hyperbolic functor H . Moreover, since BQ H ( A ) ⊂ BQ h ( A ) is the connected component of 0, we obtain from Theorem 3.2the following result Corollary 3.9.
For each i ≥ , there is an isomorphism of abelian groups GW i ( A ) ≃ GW ⊕ H,i ( A ) . It remains to compute GW ( A ) and to determine the homotopy type of theconnected components of GW ( A ) which do not contain 0. This is done on the nextsection.3.4. The connected components of GW ( A ) . We continue to denote by A a uniquely split proto-exact category with duality. In the previous section, wedetermined the homotopy groups GW i ( A ), i ≥
1, in terms of GW ⊕ H ( A ). However,we did not describe GW ( A ). As discussed in Section 3, both GW ( A ) and W ( A )are (commutative) monoids, and not groups, due to the lack of a diagonal map for ⊕ . This complicates their computation. To get around this problem, we henceforthrestrict attention to combinatorial categories.Denote by S I ⊂ S h the full subcategory of isotropically simple symmetric forms.Given S ∈ S I , we abbreviate Aut S h ( S ) to G S and write B G S for the associatedgroupoid with one object. The geometric realization of B G S is BG S . -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 23 Lemma 3.10.
Let A be a uniquely split combinatorial noetherian proto-exact cat-egory with duality.(i) The category S I is closed under direct sum. In particular, the set π ( S I ) ofconnected components of S I is a commutative monoid.(ii) The natural map π ( S I ) → W ( A ) is an isomorphism of monoids.(iii) Direct sum induces an equivalence of categories S I × S H → S h .Proof. Let S , S ∈ S I and suppose that U S ⊕ S is isotropic. The combinatorialassumption implies that U ∩ S i S i , i = 1 ,
2, is isotropic and hence each U ∩ S i isthe zero object. It follows that U ≃ S ⊕ S is isotropically simple.By definition, W ( A ) is the quotient of π ( S h ) by the submonoid of metabolic ob-jects. By Lemma 1.8, metabolics are necessarily hyperbolic. The second statementtherefore follows from Proposition 1.11.The third statement also follows from Proposition 1.11: essentially surjectivityfollows from part (i) and fully faithfulness follows from the proof of part (ii). (cid:3) The remainder of this section is devoted to proving the following theorem.
Theorem 3.11.
Let A be a uniquely split combinatorial noetherian proto-exactcategory with duality. Then there is a weak homotopy equivalence GW ( A ) ≃ G w ∈ W ( A ) BG S w × GW H ( A ) , where S w ∈ S I is an isotropically simple representative of w ∈ W ( A ) under theisomorphism of Lemma 3.10(ii). For each S ∈ S I , denote by Q S ⊂ Q h the full subcategory of objects isometric toan object of the form S ⊕ H ( V ) for some V ∈ A . Let GW S be the homotopy fibre of BF : BQ S → BQ over 0. With this notation, we have Q = Q H and GW = GW H . Lemma 3.12.
Work in the setting of Theorem 3.11.(i) For each S ∈ S I , there is an equivalence of categories Q S ≃ B G S × Q H .(ii) The category Q h decomposes into connected components as Q h = G w ∈ W ( A ) Q S w . (iii) The space GW decomposes into (not-necessarily connected) components as GW = G w ∈ W ( A ) GW S w . Proof.
Let η : BG S × Q H → Q S be the functor given on objects and morphisms by H ( V ) S ⊕ H ( V ) and (cid:18) φ, H ( V ) π և −− E i H ( V ′ ) (cid:19) (cid:18) S ⊕ H ( V ) id S ⊕ π և −−−− S ⊕ E φ ⊕ i S ⊕ H ( V ′ ) (cid:19) , respectively. We claim that η is an equivalence. Essentially surjectivity of η followsfrom the definition of Q S . Because S is isotropically simple, the combinatorialassumption ensures that any isotropic morphism with target S ⊕ H ( V ′ ) factorsthrough i H ( V ′ ) : H ( V ′ ) S ⊕ H ( V ′ ). It follows from this that η is full. A shortmatrix calculation shows that η is faithful. Consider the second statement. By Proposition 1.11, the canonical functor F w ∈ W Q S w → Q h is essentially surjective. Suppose that there is a morphism M → M in Q h . By Propositions 1.10 and 1.11, there exist isometries M ≃ S ⊕ H ( U )and M ≃ S ⊕ H ( V ) for a unique (up to isometry) S ∈ S I . It follows from Lemma3.10(ii) that M and M represent the same class in W ( A ), proving fully faithful-ness of F w ∈ W Q S w → Q h . Since S w ∈ Q S w , the category Q S w is connected.Since GW is a homotopy fibre of BQ h → BQ , the third statement follows fromthe second. (cid:3) By Lemma 3.12, in order to understand GW , it suffices to describe the compo-nents GW S . Similarly to the argument in the proof of Theorem 3.2, which treatsthe case S = 0, we can apply [13, Proposition 3.4] to the square S − E × Q Q S S − E Q S Q ˜ g ∗ y ˜ gF to see that GW S is homotopy equivalent to the geometric realization of the category S − E × Q Q S . We therefore proceed to consider the category S − E × Q Q S . Proposition 3.13.
For each S ∈ S I , there is an equivalence of categories BG S × ( S − E × Q Q H ) ≃ S − E × Q Q S . Proof.
Consider the category
E × Q Q S . By definition, an object is a triple(( A B ։ C ) ∈ E , M ∈ Q S , ( φ : C ∼ −→ M ) ∈ Q )and a morphism is a pair of morphisms in E and Q S which make the obvious squarein Q commute. We claim that there is an equivalence E × Q Q S ≃ BG S × E × Q Q H . Using Propositions 1.10 and 1.11, we see that
E × Q Q S is equivalent to the category E S whose objects are conflations of the form A S ⊕ H ⊕ A S ⊕ H, i A π S ⊕ H for some hyperbolic object H , and whose morphisms are equivalence classes ofcommutative diagrams of the form A S ⊕ H ⊕ A S ⊕ HA ′ S ⊕ H ⊕ A S ⊕ WA ′ S ⊕ H ′ ⊕ A ′ S ⊕ H ′ . i A π S ⊕ H α i A ◦ α β π id S ⊕ pφ ⊕ ii A ′ π S ′⊕ H ′ -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 25 In the previous diagram, all rows are conflations and the right column is a morphismin Q S or, by Lemma 3.12(i), in BG S × Q H . By splitting the morphism p, any suchdiagram is equivalent to one of the form A S ⊕ H ⊕ A S ⊕ HA ′ S ⊕ H ⊕ A S ⊕ H ⊕ XA ′ S ⊕ H ′ ⊕ A ′ S ⊕ H ′ i A π S ⊕ H α i A ◦ α β π π S ⊕ H φ ⊕ ii A ′ π S ⊕ H ′ (4)where W ≃ H ⊕ X . By the commutativity of the top right square, π is of the form id S ∗ ∗ x ∗ ∗ : S ⊕ H ⊕ A ։ S ⊕ H ⊕ X. By the commutativity of the bottom right square, β is of the form β = φ ∗ ∗ z ∗ ∗ : S ⊕ H ⊕ A S ⊕ H ′ ⊕ A ′ . Since φ is an isomorphism, Lemma 1.6 implies that z = 0. Commutativity of righthand squares then gives x = π X ◦ π ◦ i S = 0 and π H ⊕ X ◦ π ◦ i S = 0, with the latterequality following from the equalities( φ ⊕ i ) ◦ i H ⊕ X ◦ π H ⊕ X ◦ π ◦ i S = i H ′ ◦ π H ′ ◦ ( φ ⊕ i ) ◦ π ◦ i S = π S ⊕ H ′ ◦ β ◦ i S = 0and the fact that ( φ ⊕ i ) ◦ i H ⊕ X is a monomorphism. Hence, S decouples from thediagram (4) and the obvious functor BG S × E → E S is an equivalence. By theargument above, specialized to the case S = 0, there is an equivalence E ≃ E × Q Q H .Summarizing, we obtain an equivalence BG S × E × Q Q H ≃ E × Q Q S .Let S act on E × Q Q S by the standard action on E (see Section 2.3) and on Q S by the trivial action. Then there is an equivalence S − E × Q Q S ≃ S − ( E × Q Q S )given by simply reordering the bracketing of objects and morphisms. Hence, thereare equivalences BG S × ( S − E × Q Q H ) ≃ BG S × S − ( E × Q Q H ) ≃ S − ( E × Q Q S ) ≃ S − E × Q Q S . This completes the proof. (cid:3)
Proof of Theorem 3.11.
The space GW S is homotopy equivalent to the geometricrealization of S − E × Q Q S . By Proposition 3.13, there is an equivalence S − E × Q Q S ≃ BG S × ( S − E × Q Q H ) . Since the geometric realization of S − E × Q Q H is weak homotopy equivalent to GW H , this completes the proof. (cid:3) Corollary 3.14.
In the setting of Theorem 3.11, there is an isomorphism of monoids GW ( A ) ≃ W ( A ) × GW H, ( A ) . Proof.
By Proposition 3.13, we see that GW S ( A ) ≃ BG S × GW H ( A ) for each S ∈ S I . In particular, there is a monoid isomorphism π GW S ( A ) ≃ GW H, ( A ).In view of Theorem 3.11, it follows that the stated isomorphism holds at the levelof sets. Using Lemma 3.10(i), it is easy to see that the decomposition in Theorem3.11 is compatible with direct sum, which implies the statement for monoids. (cid:3) Using Corollary 3.14 and the discussion above Corollary 3.9 to identify π H ∗ withthe hyperbolic map, we find a monoid isomorphism W ( A ) ≃ coker( K ( A ) π H ∗ −−−→ GW ( A )) . In particular, the definition of the higher Witt groups W i ( A ), i ≥
1, given in Section3.1 extends compatibly to i = 0. Example 3.15.
We continue our running test case
Vect F from Examples 1.7, 1.12and 2.1. As shown in [8, § GW H ( Vect F ) ≃ GW ⊕ H ( Vect F ) ≃ Z × B (cid:0) ( Z / ≀ Σ ∞ (cid:1) + and GW ⊕ ( Vect F ) ≃ Z × B (cid:0)(cid:0) ( Z / ≀ Σ ∞ (cid:1) × Σ ∞ (cid:1) + and a weak homotopy equivalence GW ( Vect F ) ≃ G n ∈ Z ≥ B Σ n × Z × B (( Z / ≀ Σ ∞ ) + . This exemplifies, in particular, that the Q -construction and the +-construction leadto significantly different Grothendieck–Witt spaces for proto-exact categories. References [1] R. Charney and R. Lee. On a theorem of Giffen.
Michigan Math. J. , 33(2):169–186, 1986.[2] C. Chu, O. Lorscheid, and R. Santhanam. Sheaves and K -theory for F -schemes. Adv. Math. ,229(4):2239–2286, 2012.[3] C. Chu and J. Morava. On the algebraic K -theory of monoids. arXiv:1009.3235, 2010.[4] A. Connes and C. Consani. Schemes over F and zeta functions. Compos. Math. , 146(6):1383–1415, 2010.[5] A. Deitmar. Remarks on zeta functions and K -theory over F . Proc. Japan Acad. Ser. AMath. Sci. , 82(8):141–146, 2006.[6] A. Deitmar. Belian categories.
Far East J. Math. Sci. (FJMS) , 70(1):1–46, 2012.[7] T. Dyckerhoff and M. Kapranov.
Higher Segal Spaces . Lecture Notes in Mathematics.Springer, 2019.[8] J. Eberhardt, O. Lorscheid, and M. Young. Algebraic K -theory and Grothendieck–Witt the-ory of monoid schemes. arXiv:2009.XXXXX, 2020.[9] C. Eppolito, J. Jun, and M. Szczesny. Proto-exact categories of matroids, Hall algebras, and K -theory. Math. Z. , 296(1-2):147–167, 2020.[10] D. Grayson. Higher algebraic K -theory. II (after Daniel Quillen). In Algebraic K -theory (Proc.Conf., Northwestern Univ., Evanston, Ill., 1976) , pages 217–240. Lecture Notes in Math., Vol.551, 1976.[11] J. Hekking. Segal objects in homotopical categories & K -theory of proto-exact categories . 2017.Thesis (Master’s)–Mathematisch Instituut, Universiteit Leiden.[12] J. Hornbostel. Constructions and d´evissage in Hermitian K -theory. K -Theory , 26(2):139–170,2002.[13] J. Hornbostel and M. Schlichting. Localization in Hermitian K -theory of rings. J. LondonMath. Soc. (2) , 70(1):77–124, 2004. -THEORY AND GW -THEORY OF PROTO-EXACT CATEGORIES 27 [14] D. Quillen. Higher algebraic K -theory. I. In Algebraic K -theory, I: Higher K -theories (Proc.Conf., Battelle Memorial Inst., Seattle, Wash., 1972) , pages 85–147. Lecture Notes in Math.,Vol. 341. Springer, Berlin, 1973.[15] M. Schlichting. Hermitian K -theory on a theorem of Giffen. K -Theory , 32(3):253–267, 2004.[16] M. Schlichting. Hermitian K -theory of exact categories. J. K-Theory , 5(1):105–165, 2010.[17] M. Schlichting. Hermitian K -theory, derived equivalences and Karoubi’s fundamental theo-rem. J. Pure Appl. Algebra , 221(7):1729–1844, 2017.[18] M. Schlichting. Higher K -theory of forms I: From rings to exact categories. J. Inst. Math.Jussieu , pages 1–69, 2019.[19] V. Srinivas.
Algebraic K -theory . Modern Birkh¨auser Classics. Birkh¨auser Boston, Inc.,Boston, MA, second edition, 2008.[20] M. Uridia. U -theory of exact categories. In K -theory and homological algebra (Tbilisi, 1987–88) , volume 1437 of Lecture Notes in Math. , pages 303–313. Springer, Berlin, 1990.[21] F. Waldhausen. Algebraic K -theory of spaces. In Algebraic and geometric topology (NewBrunswick, N.J., 1983) , volume 1126 of
Lecture Notes in Math. , pages 318–419. Springer,Berlin, 1985.[22] C. Weibel.
The K -book: An introduction to algebraic K -theory , volume 145 of GraduateStudies in Mathematics . American Mathematical Society, Providence, RI, 2013.[23] M. Young. Relative 2-Segal spaces.
Algebr. Geom. Topol. , 18(2):975–1039, 2018.[24] M. Young. Degenerate versions of Green’s theorem for Hall modules.
J. Pure Appl. Algebra ,225(4):106557, 2021.
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