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[MATH] for any [MATH] such that [EQUATION] If we choose [EQUATION] and define [MATH] , then [MATH] Let [MATH] be the cokernel of the inclusion [MATH] Then [MATH] is lean/split and insular and has a grading given by [MATH] . Since |
[EQUATION] the quotient [MATH] is in [MATH] , and [MATH] factors as [MATH] in the right square in the map of exact sequences [EQUATION] |
as required. 3.5.7 Definition The category [MATH] is the localization [MATH] It is clear that the quotient [MATH] is an additive category, and [MATH] is an additive functor. In fact, we have the following. |
3.5.8 Theorem The short sequences in [MATH] which are isomorphic to images of exact sequences from [MATH] form a Quillen exact structure. |
Proof. This will be a consequence from 17 , Proposition 1.16] Since [MATH] is right filtering by Lemma 3.5.6 , it remains to check that [MATH] right s-filtering in [MATH] in the following sense. A subcategory [MATH] of an exact category [MATH] is right s-filtering if given an admissible monomorphism [MATH] with [MATH] ... |
Suppose that [MATH] and [MATH] have the same properties as in the proof of Lemma 3.5.6 , and [MATH] Since [MATH] is in [MATH] , there are [MATH] and a monotone function |
[MATH] such that [MATH] Then let [MATH] where [EQUATION] Define [MATH] as the cokernel of the inclusion [MATH] and let [MATH] be the quotient map. The composition [MATH] is an admissible monomorphism with [MATH] |
3.5.9 Notation If [MATH] is a subset of [MATH] as before, [MATH] will stand for the exact category [MATH] and [MATH] for its Quillen [MATH] -theory. |
The main tool in proving controlled excision theorems will be the following localization sequence. 3.5.10 Theorem (Theorem 2.1 of Schlichting |
Let [MATH] be an idempotent complete right s-filtering subcategory of an exact category [MATH] which is full and closed under exact extensions. Then the sequence of exact categories |
[MATH] induces a homotopy fibration of Quillen [MATH] -theory spectra [EQUATION] 3.5.11 Corollary There is a homotopy fibration [EQUATION] |
There is a more intrinsic statement of the same fact. 3.5.12 Theorem (Localization) There is a homotopy fibration [EQUATION] Theorem 3.5.12 is a consequence of Corollary 3.5.11 |
as soon as we show that [MATH] and [MATH] are weakly equivalent. Recall that the essential full image of a functor [MATH] is the full subcategory of [MATH] whose objects are those [MATH] that are isomorphic to [MATH] for some [MATH] from [MATH] |
3.5.13 Lemma Given a pair of proper metric spaces [MATH] , there is a fully faithful embedding [MATH] The Serre subcategory [MATH] is the essential full image of [MATH] in [MATH] Therefore, the inclusion [MATH] induces a weak equivalence |
[EQUATION] Proof. Suppose [MATH] is an object of [MATH] The embedding [MATH] is given by [MATH] [MATH] It is clear that [MATH] is in [MATH] |
To show that [MATH] is the essential full image, for an object [MATH] of [MATH] assume that [MATH] for some number [MATH] and a monotone function [MATH] Choose any set function |
[MATH] with the properties (1) [MATH] (2) [MATH] for all [MATH] in [MATH] (3) [MATH] Then the [MATH] -filtered module [MATH] associated to [MATH] given by [MATH] with the grading |
[MATH] is an object of [MATH] Indeed, if [MATH] is [MATH] -lean/split and [MATH] -insular then [MATH] is [MATH] -lean/split and [MATH] -insular. The identity map is an isomorphism in [MATH] with [MATH] |
4. Fibrewise excision theorems 4.1. Waldhausen categories and -theory Our main reference for Waldhausen [MATH] -theory terminology and notation is Thomason |
A Waldhausen category [MATH] with weak equivalences [MATH] is often denoted by [MATH] as a reminder of the choice. A functor between Waldhausen categories is exact if it preserves the chosen zero objects, cofibrations, weak equivalences, and cobase changes. Let [MATH] be a small Waldhausen category with respect to two ... |
[MATH] satisfies the cylinder axiom then the induced functor does so too. 4.1.1 Theorem (Approximation Theorem) Let [MATH] be an exact functor between two small saturated Waldhausen categories. It induces a map of |
[MATH] -theory spectra [EQUATION] Assume that [MATH] has a cylinder functor satisfying the cylinder axiom. If [MATH] satisfies two conditions: |
(1) a morphism [MATH] is in [MATH] if and only if [MATH] is in [MATH] (2) for any object [MATH] and any morphism [MATH] in [MATH] , there is an object [MATH] , a morphism [MATH] in [MATH] , and a weak equivalence [MATH] such that [MATH] |
then [MATH] is a homotopy equivalence. Proof. This is Theorem 1.6.7 of . The presence of the cylinder functor with the cylinder axiom allows to make condition (2) weaker than that of Waldhausen, see point 1.9.1 in |
4.1.2 Definition In any additive category, a sequence of morphisms [EQUATION] is called a (bounded) chain complex if the compositions |
[MATH] are the zero maps for all [MATH] ,…, [MATH] . A chain map [MATH] is a collection of morphisms [MATH] such that [MATH] . A chain map [MATH] is null-homotopic if there are morphisms [MATH] such that [MATH] . Two chain maps [MATH] [MATH] |
are chain homotopic if [MATH] is null-homotopic. Now [MATH] is a chain homotopy equivalence if there is a chain map [MATH] such that the compositions [MATH] and [MATH] |
are chain homotopic to the respective identity maps. The Waldhausen structures on categories of bounded chain complexes are based on homotopy equivalence as a weakening of the notion of isomorphism of chain complexes. |
A sequence of maps in an exact category is called acyclic if it is assembled out of short exact sequences in the sense that each map factors as the composition of the cokernel of the preceding map and the kernel of the succeeding map. |
It is known that the class of acyclic complexes in an exact category is closed under isomorphisms in the homotopy category if and only if the category is idempotent complete, which is also equivalent to the property that each contractible chain complex is acyclic, cf. 12 , sec. 11] |
Given an exact category [MATH] , there is a standard choice for the Waldhausen structure on the category [MATH] of bounded chain complexes in [MATH] where the degree-wise admissible monomorphisms are the cofibrations and the chain maps whose mapping cones are homotopy equivalent to acyclic complexes are the weak equiva... |
The following fact is well-known, cf. point 1.1.2 in 4.1.3 Proposition The category [MATH] is a Waldhausen category satisfying the extension and saturation axioms and has cylinder functor satisfying the cylinder axiom. |
4.1.4 Example There are two choices for the Waldhausen structure on the category of bounded chain complexes [MATH] . One is the standard choice [MATH] as above. Given a subset [MATH] another choice for the weak equivalences [MATH] is the chain maps whose mapping cones are homotopy equivalent to acyclic complexes in the... |
4.1.5 Corollary The categories [MATH] and [MATH] are Waldhausen categories satisfying the extension and saturation axioms and have cylinder functors satisfying the cylinder axiom. |
Proof. All axioms and constructions, including the cylinder functor, for [MATH] are inherited from [MATH] The [MATH] -theory functor from the category of small Waldhausen categories [MATH] and exact functors to the category of connective spectra is defined in terms of [MATH] -construction as in Waldhausen |
. It extends to simplicial categories [MATH] with cofibrations and weak equivalences and inductively delivers the connective spectrum [MATH] . We obtain the functor assigning to [MATH] the connective |
[MATH] -spectrum [EQUATION] representing the Waldhausen algebraic [MATH] -theory of [MATH] . For example, if [MATH] is the additive category of free finitely generated [MATH] -modules with the canonical Waldhausen structure, then the stable homotopy groups of [MATH] are the usual |
[MATH] -groups of the ring [MATH] . In fact, there is a general identification of the two theories. Recall that for any exact category [MATH] , the category [MATH] of bounded chain complexes has the Waldhausen structure [MATH] as in Example 4.1.4 |
4.1.6 Theorem The Quillen [MATH] -theory of an exact category [MATH] is equivalent to the Waldhausen [MATH] -theory of [MATH] Proof. |
The proof is based on repeated applications of the Additivity Theorem, cf. Thomason’s Theorem 1.11.7 from . Thomason’s proof of his Theorem 1.11.7 can be repeated verbatim here. It is in fact simpler in this case since his condition 1.11.3.1 is not required. |
4.2. Controlled excision theorems These are the major computational tools in controlled [MATH] -theory. We develop excision results [MATH] with respect to specific coverings of the variable [MATH] in this section. |
Suppose [MATH] and [MATH] are subsets of a proper metric space [MATH] , and [MATH] We use the notation [MATH] [MATH] for [MATH] or [MATH] , and [MATH] for the intersection |
[MATH] Theorem 3.5.10 can be applied to two inclusions of categories, [MATH] and [MATH] . The resulting homotopy fibrations are the rows in a commutative diagram of [MATH] -theory spectra |
[EQUATION] The vertical maps are induced by exact inclusions, including [MATH] induced by the exact functor [MATH] which is itself induced from the exact inclusion [MATH] |
4.2.1 Remark This is precisely the commutative diagram from Cardenas-Pedersen , section 8] transported from bounded [MATH] -theory to fibred [MATH] -theory. Cardenas and Pedersen use Karoubi quotients and the Karoubi fibrations in order to generate their diagram. One of the crucial points in |
is that the functor [MATH] between the Karoubi quotients is an isomorphism of categories. In fibred [MATH] -theory the situation is more complicated: [MATH] is not necessarily full and, therefore, not an isomorphism of categories. We will use the Approximation Theorem to prove that [MATH] is nevertheless an equivalence... |
4.2.2 Proposition [MATH] Proof. This follows from Lemma 2.3 in as part of the proof of Theorem 3.5.10 where [MATH] from Waldhausen’s Fibration Theorem is identified with the Quillen [MATH] -theory spectrum [MATH] |
4.2.3 Lemma If [MATH] is a degreewise admissible monomorphism with cokernel in [MATH] then [MATH] is a weak equivalence in [MATH] |
Proof. The mapping cone [MATH] is quasi-isomorphic to the cokernel of [MATH] , by Lemma 11.6 of , which is zero in [MATH] The exact inclusion [MATH] induces the exact functor [MATH] |
4.2.4 Lemma The map [MATH] is a weak equivalence. Proof. Applying the Approximation Theorem, condition (1) is clear, so we need to check condition (2). Consider |
[EQUATION] in [MATH] and a chain map [MATH] for some complex [EQUATION] in [MATH] Suppose all [MATH] and [MATH] are [MATH] -lean/split and [MATH] -insular. Also assume that there is a fixed number [MATH] and a monotone function [MATH] such that |
[MATH] holds for all [MATH] If the pair [MATH] serves as bounded control data for all [MATH] [MATH] , and [MATH] , we define the submodule |
[EQUATION] and define [MATH] to be the restrictions of [MATH] to [MATH] . This gives a chain subcomplex [MATH] of [MATH] in [MATH] with the inclusion [MATH] . Notice that we have the induced chain map [MATH] in [MATH] so that [MATH] |
Once we establish that [MATH] is in [MATH] [MATH] is a weak equivalence by Lemma 4.2.3 Since [EQUATION] each [MATH] is supported on |
[EQUATION] cf. Lemma 3.5.6 So the complex [MATH] is indeed in [MATH] The excision theorems are best stated in terms of non-connective deloopings of the [MATH] -theory spectra. Following Pedersen and Weibel we can use the same kind of diagram to first deloop [MATH] -theory and then reuse it to prove the excision theorem... |
Let [MATH] [MATH] , and [MATH] denote the metric spaces of the reals, the nonnegative reals, and the nonpositive reals with the restriction of the usual metric on the real line [MATH] . Then there is the following instance of commutative diagram ( [MATH] |
[EQUATION] We already know that [MATH] is an equivalence. 4.2.5 Lemma The spectra [MATH] and [MATH] are contractible. Proof. This follows from the fact that these controlled categories are flasque, that is, the evident shift functor [MATH] in the positive (respectively negative) direction along [MATH] (respectively [MA... |
In view of Lemma 4.2.4 , we obtain a map [MATH] which induces isomorphisms of [MATH] -groups in positive dimensions. Weak equivalences |
[EQUATION] are obtained by iterating this construction for [MATH] 4.2.6 Definition The nonconnective fibred bounded [MATH] -theory over the pair [MATH] is the spectrum |
[EQUATION] Since [MATH] can be identified with [MATH] , this definition also gives a nonconnective delooping of the [MATH] -theory of [MATH] |
[EQUATION] The subcategory [MATH] is evidently a Serre subcategory of [MATH] for any choice of the subset [MATH] 4.2.7 Definition |
We define [EQUATION] We also define [EQUATION] 4.2.8 Theorem (Fibrewise Bounded Excision, Version One) Suppose [MATH] and [MATH] are subsets of a metric space [MATH] , and [MATH] There is a homotopy pushout diagram of spectra |
[EQUATION] where the maps of spectra are induced from the exact inclusions. If [MATH] and [MATH] are mutually antithetic subsets of [MATH] there is a homotopy pushout |
[EQUATION] Proof. Let us write [MATH] for [MATH] whenever [MATH] is the fibred bounded category for a pair [MATH] If [MATH] represents a family of coarsely equivalent subsets in a coarse covering [MATH] of [MATH] , consider the fibration |
[EQUATION] from Theorem 3.5.12 . Notice that there is a map [MATH] which is an equivalence in positive dimensions by the Five Lemma. Defining |
[EQUATION] gives an induced fibration [EQUATION] The theorem follows from the commutative diagram [EQUATION] and the fact that [MATH] is a weak equivalence. |
From Lemma 3.5.13 one has a weak equivalence [MATH] If [MATH] and [MATH] are coarsely antithetic then the same construction shows that [MATH] is onto the essential full image, and so there is a weak equivalence |
[MATH] This allows to substitute the terms in the commutative diagram, giving the second statement. 4.3. Fibred coarse coverings |
To state the excision theorems properly in the coarse geometric setting, we develop the language of fibred coarse coverings. Two subsets [MATH] [MATH] of [MATH] are called coarsely equivalent if there is a set of enlargement data [MATH] such that |
[MATH] and [MATH] We will use the notation [MATH] for this equivalence relation. A family of subsets [MATH] is called coarsely saturated if it is maximal with respect to this equivalence relation. Given a subset [MATH] , we denote by [MATH] the smallest boundedly saturated family containing [MATH] |
A collection of subsets [MATH] is a coarse covering of [MATH] if [MATH] for some [MATH] Similarly, [MATH] is a coarse covering by coarsely saturated families if for some (and therefore any) choice of subsets [MATH] [MATH] is a coarse covering in the above sense. |
We will say that a pair of subsets [MATH] [MATH] of [MATH] are coarsely antithetic if for any two sets of enlargement data [MATH] and [MATH] there exist enlargement data [MATH] such that |
[EQUATION] We will write [MATH] to indicate that [MATH] and [MATH] are coarsely antithetic. Given two subsets [MATH] and [MATH] , we define |
[EQUATION] It is easy to see that [MATH] is a coarsely saturated family. 4.3.1 Proposition [MATH] is a coarsely saturated family. |
Proof. Suppose [MATH] [MATH] and [MATH] [MATH] are two coarsely antithetic pairs, and [MATH] [MATH] for some [MATH] [MATH] [MATH] , and [MATH] Then |
[EQUATION] for some [MATH] There is the straightforward generalization to the case of a finite number of subsets of [MATH] Similarly, we write [MATH] if for arbitrary sets of data [MATH] there is a set of enlargement data [MATH] so that |
[EQUATION] and define [EQUATION] Identifying any coarsely saturated family [MATH] with [MATH] for [MATH] , one has the coarse saturated family |
[MATH] We will refer to [MATH] as the coarse intersection of [MATH] A coarse covering [MATH] is closed under coarse intersections if all coarse intersections [MATH] are nonempty and are contained in [MATH] If [MATH] is a given coarse covering, the smallest coarse covering that is closed under coarse intersections and c... |
[MATH] will be called the closure of [MATH] under coarse intersections. All of the terms introduced above have absolute analogues obtained by simply restricting to the case [MATH] . So there are, in particular, finite coarse coverings of a single metric space. |
4.3.2 Proposition If [MATH] is a finite coarse antithetic covering of [MATH] then [MATH] consisting of subsets [MATH] [MATH] , is a coarse antithetic covering of [MATH] . If [MATH] is closed under coarse intersections, [MATH] is closed under coarse intersections. |
Proof. Suppose [MATH] so that for [MATH] [MATH] is a coarse covering of [MATH] . Then [MATH] is a covering of [MATH] . Suppose [MATH] is coarsely antithetic, so given numbers [MATH] [MATH] there is a number [MATH] so that [MATH] . If [MATH] [MATH] are non-decreasing functions, these values give a non-decreasing functio... |
[EQUATION] where [MATH] can be any non-negative number, and [MATH] is the function [MATH] . So [MATH] is a coarsely antithetic covering. A similar estimate gives the last statement. |
Suppose [MATH] is a finite coarse covering of [MATH] closed under coarse intersections. We can define the homotopy pushout [EQUATION] |
4.3.3 Theorem (Fibrewise Bounded Excision, Version Two) There is a weak equivalence [EQUATION] Proof. Apply Theorem 4.2.8 inductively to the sets in [MATH] |
4.4. Relative excision theorems Fibred [MATH] -theory has a useful relative version, and there are generalizations of the excision theorems to relative statements. |
4.4.1 Definition Let [MATH] for a coarse covering [MATH] of [MATH] . Let [MATH] and [MATH] . The category [MATH] is the quotient category [MATH] |
It is now straightforward to define [EQUATION] [EQUATION] and [EQUATION] The theory developed in this section is spontaneously relativized to give the following excision theorem. |
4.4.2 Theorem (Relative Fibrewise Excision, Version One) If [MATH] is the union of two subsets [MATH] and [MATH] there is a homotopy pushout diagram of spectra |
[EQUATION] where the maps of spectra are induced from the exact inclusions. In fact, if [MATH] is the union of two mutually antithetic subsets [MATH] and [MATH] , and [MATH] is antithetic to both [MATH] and [MATH] there is a homotopy pushout |
[EQUATION] Finally, we want to state the relative excision theorem in the most familiar form. 4.4.3 Proposition Given a subset [MATH] of [MATH] , there is a weak equivalence |
[EQUATION] Proof. Consider the setup of Theorem 4.2.8 with [MATH] and [MATH] , then Lemma 4.2.4 shows that the map [EQUATION] induces a weak equivalence on the level of [MATH] -theory. |
Notice that, since [MATH] is a subset of [MATH] , we have the interpretation [EQUATION] Now the maps of quotients [EQUATION] and |
[EQUATION] induced by fully faithful embeddings also induce weak equivalences. Their composition gives the required equivalence. |
The relative theorem can be restated using coarse coverings in terms of the homotopy pushout [EQUATION] 4.4.4 Theorem (Relative Fibrewise Excision, Version Two) |
There is a weak equivalence [EQUATION] Proof. Apply Theorem 4.4.2 inductively to the sets in [MATH] 5. Conclusion It is a familiar fact that [MATH] -theoretic approximations to the usually more meaningful [MATH] -theoretic invariants are easier to compute. This paper confirms the pattern in the controlled algebra setti... |
and perform computations in more general geometric settings. This material will appear in , while the relationship between the [MATH] -theory of group rings for finitely generated groups and a [MATH] -theoretic analogue based on controlled [MATH] -theory is studied in |
. For the purpose of stating the results we restrict to regular coefficient rings [MATH] of finite global dimension. The conclusion is that the appropriate [MATH] -theory of the group ring is computable leveraging the results of this paper, while the Cartan comparison map from the [MATH] -theory is an equivalence for a... |
# Source: arxiv 1806.08787 # Title: A proposed solution for analysis management in high energy physics # Sections: all # Downloaded: 2026-03-03T05:16:35.453877+00:00 |
A proposed solution for analysis management in high energy physics \skipbeforetitle 100pt \abstracttext This paper presents an architecture for the analysis management in high energy physics experiments. Some new concepts on data analysis are introduced. A protocol for organizing and operating an analysis is raised. A ... |
Introduction A physics analysis should be well managed. One of the most important aspects in analysis management is the analysis preservation. Analysis preservation is essential in scientific research. It is the responsibility of a researcher to clarify the details to produce the result from the raw data. The details i... |
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