STRONG HOMOTOPY TYPES, NERVES AND COLLAPSES 1
... of this subject, since we are more interested in understanding the difference between the various notions of collapses from a geometric point of view. One of the most significant questions related to this concept is the so-called Evasiveness conjecture for simplicial complexes which asserts that a n ...
... of this subject, since we are more interested in understanding the difference between the various notions of collapses from a geometric point of view. One of the most significant questions related to this concept is the so-called Evasiveness conjecture for simplicial complexes which asserts that a n ...
Homotopy theories and model categories
... pushouts and homotopy pullbacks in a model category. What we do along these lines can certainly be carried further. This paper is not in any sense a survey of everything that is known about model categories; in fact we cover only a fraction of the material in [22]. The last section has a discussion ...
... pushouts and homotopy pullbacks in a model category. What we do along these lines can certainly be carried further. This paper is not in any sense a survey of everything that is known about model categories; in fact we cover only a fraction of the material in [22]. The last section has a discussion ...
Homotopy Theory of Topological Spaces and Simplicial Sets
... objects will have some common properties, depending on the category. For example in the setting of sets as before isomorphisms are just bijections and the sets that are isomorphic will have the same cardinality. The isomorphic objects in the homotopy category that we are going to construct will be ’ ...
... objects will have some common properties, depending on the category. For example in the setting of sets as before isomorphisms are just bijections and the sets that are isomorphic will have the same cardinality. The isomorphic objects in the homotopy category that we are going to construct will be ’ ...
a note on trivial fibrations - Fakulteta za matematiko in fiziko
... It is well-known that every fibration over a contractible base is fibre homotopically trivial. In fact, it is easy to see that this property characterizes contractible spaces. It is less clear whether a fibration with contractible fibre must also be fibre homotopically trivial, and whether the trivi ...
... It is well-known that every fibration over a contractible base is fibre homotopically trivial. In fact, it is easy to see that this property characterizes contractible spaces. It is less clear whether a fibration with contractible fibre must also be fibre homotopically trivial, and whether the trivi ...
Tibor Macko
... Theorem 1.1 ([Ran92], Theorem 17.4). Let X be a finite Poincaré complex of formal dimension n ≥ 5. Then X is homotopy equivalent to a closed n-dimensional topological manifold if and only if 0 = s(X) ∈ Sn (X). By choosing a simplicial complex homotopy equivalent to X we can assume that X is a simpl ...
... Theorem 1.1 ([Ran92], Theorem 17.4). Let X be a finite Poincaré complex of formal dimension n ≥ 5. Then X is homotopy equivalent to a closed n-dimensional topological manifold if and only if 0 = s(X) ∈ Sn (X). By choosing a simplicial complex homotopy equivalent to X we can assume that X is a simpl ...
FIBRATIONS AND HOMOTOPY COLIMITS OF
... must preserve such if p : ShB → E is a boolean localization of E. 3.15. Model category for simplicial presheaves. Although we will not make much use of it here, we note that if E = ShC for some Grothendieck site C, then Jardine [6, Thm. 17] constructs a “presheaf” closed model category structure on ...
... must preserve such if p : ShB → E is a boolean localization of E. 3.15. Model category for simplicial presheaves. Although we will not make much use of it here, we note that if E = ShC for some Grothendieck site C, then Jardine [6, Thm. 17] constructs a “presheaf” closed model category structure on ...
Non-commutative Donaldson--Thomas theory and vertex operators
... spaces admit symmetric obstruction theory and the invariants are defined as the virtual counting of the moduli spaces in the sense of Behrend–Fantechi [3]4 . In the case when C D D ∅, we show that the moduli space MncDT .; ∅; / admits a symmetric obstruction theory (Section 5.2). Using the resu ...
... spaces admit symmetric obstruction theory and the invariants are defined as the virtual counting of the moduli spaces in the sense of Behrend–Fantechi [3]4 . In the case when C D D ∅, we show that the moduli space MncDT .; ∅; / admits a symmetric obstruction theory (Section 5.2). Using the resu ...
RATIONAL HOMOTOPY THEORY Contents 1. Introduction 1 2
... You might again hope something like Hn ( X ) = Hn ( X1 ) ⊕ Hn ( X2 /X1 ). Of course, this is not true. What IS true is that there is an induced filtration on homology: ...
... You might again hope something like Hn ( X ) = Hn ( X1 ) ⊕ Hn ( X2 /X1 ). Of course, this is not true. What IS true is that there is an induced filtration on homology: ...
HOMOLOGICAL PROPERTIES OF NON
... By now, this homological result does not have any known computer scientific interpretation. But it sheds some light on the potential of an algebraic topological approach of concurrency. At last, Section 9 then gives several examples of calculation which illustrate the mathematical notions presented ...
... By now, this homological result does not have any known computer scientific interpretation. But it sheds some light on the potential of an algebraic topological approach of concurrency. At last, Section 9 then gives several examples of calculation which illustrate the mathematical notions presented ...
Homotopy theory for beginners - Institut for Matematiske Fag
... If two maps are homotopic rel A, then they are homotopic. A pointed space is a pair (X, x0 ) consisting of space X and one of its points x0 ∈ X. The pointed topological category, Top∗ , is the category where the objects are pointed topological spaces and the morphisms are base-point preserving maps, ...
... If two maps are homotopic rel A, then they are homotopic. A pointed space is a pair (X, x0 ) consisting of space X and one of its points x0 ∈ X. The pointed topological category, Top∗ , is the category where the objects are pointed topological spaces and the morphisms are base-point preserving maps, ...
A Prelude to Obstruction Theory - WVU Math Department
... Let us now apply mathematical rigor to our idea of a CW-complex. Definition 1.3.1. Recall that a space holds the T2 separation axiom, or is Hausdorff, if, for any two elements of the space, there are mutually disjoint neighborhoods of these points. That is to say, for each x, y ∈ X, there exists a n ...
... Let us now apply mathematical rigor to our idea of a CW-complex. Definition 1.3.1. Recall that a space holds the T2 separation axiom, or is Hausdorff, if, for any two elements of the space, there are mutually disjoint neighborhoods of these points. That is to say, for each x, y ∈ X, there exists a n ...
AN INTRODUCTION TO ∞-CATEGORIES Contents 1. Introduction 1
... of a homotopical category, like model structures, enriching over spaces, or Segal spaces. But the question is: Can we write a Maclane-type tome of results using those other frameworks? No doubt we one day will, but we haven’t. The only setting in which all the useful results from classical category ...
... of a homotopical category, like model structures, enriching over spaces, or Segal spaces. But the question is: Can we write a Maclane-type tome of results using those other frameworks? No doubt we one day will, but we haven’t. The only setting in which all the useful results from classical category ...
1 The fundamental group of a topological space
... Given a topology T on a set Z, one says that S ⊂ T is a prebasis of T if any element of T may be expressed as an arbitrary union of finite intersections of elements of S (in this case, the family of finite intersections of elements of S is said to be a basis of T ). Any S ⊂ P(Z) generates in such a ...
... Given a topology T on a set Z, one says that S ⊂ T is a prebasis of T if any element of T may be expressed as an arbitrary union of finite intersections of elements of S (in this case, the family of finite intersections of elements of S is said to be a basis of T ). Any S ⊂ P(Z) generates in such a ...
Structure theory of manifolds
... BTop(n) and Topn = Top(n). There is a subgroup P Ln of the P L homeomorphisms in Topn . But it is an open question whether P Ln can be given the structure of a topological group in such a way that BP Ln = BP L(n) and hence P Ln = P L(n). We let Gn be the space of homotopy equivalences of S n−1 with ...
... BTop(n) and Topn = Top(n). There is a subgroup P Ln of the P L homeomorphisms in Topn . But it is an open question whether P Ln can be given the structure of a topological group in such a way that BP Ln = BP L(n) and hence P Ln = P L(n). We let Gn be the space of homotopy equivalences of S n−1 with ...
APPLICATIONS OF NIELSEN THEORY TO DYNAMICS
... Homotopy invariance. Suppose f ' f 0 : X → X via a homotopy {ft }0≤t≤1 . The homotopy gives rise to a homotopy equivalence Tf ' Tf 0 in a standard way. If we identify Γ0 = π1 (Tf 0 ) with Γ = π1 (Tf ) via this homotopy equivalence, then LΓ0 (f 0n ) = LΓ (f n ) for all n, hence also NΓ0 (f 0n ) = NΓ ...
... Homotopy invariance. Suppose f ' f 0 : X → X via a homotopy {ft }0≤t≤1 . The homotopy gives rise to a homotopy equivalence Tf ' Tf 0 in a standard way. If we identify Γ0 = π1 (Tf 0 ) with Γ = π1 (Tf ) via this homotopy equivalence, then LΓ0 (f 0n ) = LΓ (f n ) for all n, hence also NΓ0 (f 0n ) = NΓ ...
Structural Types for the Factorisation Calculus
... normalising [9], so we say the type system itself is strongly normalising. Hence, the Curry type system is restrictive enough to only type ‘terminating programs’. More expressive type systems, such as the polymorphic System F (section 1.4), type a larger class of terms while retaining this property. ...
... normalising [9], so we say the type system itself is strongly normalising. Hence, the Curry type system is restrictive enough to only type ‘terminating programs’. More expressive type systems, such as the polymorphic System F (section 1.4), type a larger class of terms while retaining this property. ...
Introduction to higher homotopy groups and
... groups and their uses, and at the same time to prepare a bit for the study of characteristic classes which will come next. These notes are not intended to be a comprehensive reference (most of this material is covered in much greater depth and generality in a number of standard texts), but rather to ...
... groups and their uses, and at the same time to prepare a bit for the study of characteristic classes which will come next. These notes are not intended to be a comprehensive reference (most of this material is covered in much greater depth and generality in a number of standard texts), but rather to ...
Lecture Notes on Stability Theory
... i.e. Boolean combinations of algebraic sets. Th (M) is axiomatized as the theory of algebraically closed fields of char 0, denoted ACF0 . Note in particular that every definable subset of M is either finite, or cofinite. Theories satisfying this property are called strongly minimal. (2) Let M = (R, ...
... i.e. Boolean combinations of algebraic sets. Th (M) is axiomatized as the theory of algebraically closed fields of char 0, denoted ACF0 . Note in particular that every definable subset of M is either finite, or cofinite. Theories satisfying this property are called strongly minimal. (2) Let M = (R, ...
The Development of Categorical Logic
... known as “toposophy”— whose chief tenet is the idea that, like a model of set theory, any topos may be taken as a taken as an autonomous universe of discourse or “world” in which mathematical concepts can be interpreted and constructions performed. Accordingly topos theorists sought to “relativize”— ...
... known as “toposophy”— whose chief tenet is the idea that, like a model of set theory, any topos may be taken as a taken as an autonomous universe of discourse or “world” in which mathematical concepts can be interpreted and constructions performed. Accordingly topos theorists sought to “relativize”— ...
SimpCxes.pdf
... for most purposes of basic algebraic topology. There are more general classes of spaces, in particular the finite CW complexes, that are more central to the modern development of the subject, but they give exactly the same collection of homotopy types. The relevant background on simplicial complexes ...
... for most purposes of basic algebraic topology. There are more general classes of spaces, in particular the finite CW complexes, that are more central to the modern development of the subject, but they give exactly the same collection of homotopy types. The relevant background on simplicial complexes ...
HIGHER CATEGORIES 1. Introduction. Categories and simplicial
... object in sets is called a simplicial set. The category of simplicial sets will be denoted sSet. In other words, sSet = P (∆). The category of simplicial sets is the one where most of the homotopy theory lives. Let us describe in more detail what is a simplicial set. To each [n] it assigns a set Xn ...
... object in sets is called a simplicial set. The category of simplicial sets will be denoted sSet. In other words, sSet = P (∆). The category of simplicial sets is the one where most of the homotopy theory lives. Let us describe in more detail what is a simplicial set. To each [n] it assigns a set Xn ...
Homotopy type theory
In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intensional type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.This includes, among other lines of work, the construction of homotopical and higher-categorical models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and higher category theory; the development of mathematics within a type-theoretic foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the formalization of each of these in computer proof assistants.There is a large overlap between the work referred to as homotopy type theory, and as the univalent foundations project. Although neither is precisely delineated, and the terms are sometimes used interchangeably, the choice of usage also sometimes corresponds to differences in viewpoint and emphasis. As such, this article may not represent the views of all researchers in the fields equally.