THE HOMOMORPHISMS OF TOPOLOGICAL GROUPOIDS 1
... e corresponding to an object x ∈ G0 is called the unity or identity corresponding to x. We have also following notations about the groupoids. We denote the set of arrows started at any object x ∈ G0 by Gx (or StG x), and the set of arrows ended at any object y ∈ G0 by Gy (or CostG y) in a groupoid ( ...
... e corresponding to an object x ∈ G0 is called the unity or identity corresponding to x. We have also following notations about the groupoids. We denote the set of arrows started at any object x ∈ G0 by Gx (or StG x), and the set of arrows ended at any object y ∈ G0 by Gy (or CostG y) in a groupoid ( ...
Sheaf Theory (London Mathematical Society Lecture Note Series)
... Sheaf theory provides a language for the discussion of geometric objects of many different kinds. At present it finds its main applications in topology and (more especially) in modern algebraic geometry, where it has been used with great success as a tool in the solution of several longstanding prob ...
... Sheaf theory provides a language for the discussion of geometric objects of many different kinds. At present it finds its main applications in topology and (more especially) in modern algebraic geometry, where it has been used with great success as a tool in the solution of several longstanding prob ...
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... be an introduction to K-theory, both algebraic and topological, with emphasis on their interconnections. While a wide range of topics is covered, an effort has been made to keep the exposition as elementary and self-contained as possible. Since its beginning in the celebrated work of Grothendieck on ...
... be an introduction to K-theory, both algebraic and topological, with emphasis on their interconnections. While a wide range of topics is covered, an effort has been made to keep the exposition as elementary and self-contained as possible. Since its beginning in the celebrated work of Grothendieck on ...
Finite spaces and larger contexts JP May
... Introduction A finite space is a topological space that has only finitely many points. At first glance, it seems ludicrous to think that such spaces can be of any interest. In fact, from the point of view of homotopy theory, they are equivalent to finite simplicial complexes. Therefore they support ...
... Introduction A finite space is a topological space that has only finitely many points. At first glance, it seems ludicrous to think that such spaces can be of any interest. In fact, from the point of view of homotopy theory, they are equivalent to finite simplicial complexes. Therefore they support ...
“Research Note” TOPOLOGICAL RING
... A topological groupoid is a groupoid R such that the sets R and R0 are topological spaces, and source, target, object, inverse and composition maps are continuous. Let R and H be two topological groupoids. A morphism of topological groupoids is a pair of maps f:H→R and f0:H0→R0 such that f and f0 ar ...
... A topological groupoid is a groupoid R such that the sets R and R0 are topological spaces, and source, target, object, inverse and composition maps are continuous. Let R and H be two topological groupoids. A morphism of topological groupoids is a pair of maps f:H→R and f0:H0→R0 such that f and f0 ar ...
On Chains in H-Closed Topological Pospaces
... and ↓a are closed subsets of X for each a ∈ X. A topological space equipped with a continuous partial order is called a topological partially ordered space or shortly topological pospace. A partial order on a topological space X is continuous if and only if the graph of is a closed subset in X × ...
... and ↓a are closed subsets of X for each a ∈ X. A topological space equipped with a continuous partial order is called a topological partially ordered space or shortly topological pospace. A partial order on a topological space X is continuous if and only if the graph of is a closed subset in X × ...
The Simplicial Lusternik
... the simplicial complexes L such that K & L, of the smallest number of collapsible subcomplexes that can cover L. However, the concept of collapsibility presents some difficulties, for example the core of a simplicial complex is not unique and a simplicial complex can collapse to two non-isomorphic m ...
... the simplicial complexes L such that K & L, of the smallest number of collapsible subcomplexes that can cover L. However, the concept of collapsibility presents some difficulties, for example the core of a simplicial complex is not unique and a simplicial complex can collapse to two non-isomorphic m ...
Surveys on Surgery Theory : Volume 1 Papers dedicated to C. T. C.
... called an h-cobordism if the inclusions M ,→ W and N ,→ W are homotopy equivalences. The importance of this notion stems from the h-cobordism theorem of Smale (ca. 1960), which showed that if M and N are simply connected and of dimension ≥ 5, then every h-cobordism between M and N is a cylinder M × ...
... called an h-cobordism if the inclusions M ,→ W and N ,→ W are homotopy equivalences. The importance of this notion stems from the h-cobordism theorem of Smale (ca. 1960), which showed that if M and N are simply connected and of dimension ≥ 5, then every h-cobordism between M and N is a cylinder M × ...
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... can achieve transversality only by perturbing f by a multivalued section s. Hence the virtual moduli cycle, which is defined to be the zero set of f + s, is a weighted branched submanifold of the infinite dimensional groupoid E: see [10, Ch 7]. Since s is chosen so that f + s is Fredholm (in the lan ...
... can achieve transversality only by perturbing f by a multivalued section s. Hence the virtual moduli cycle, which is defined to be the zero set of f + s, is a weighted branched submanifold of the infinite dimensional groupoid E: see [10, Ch 7]. Since s is chosen so that f + s is Fredholm (in the lan ...
homotopy types of topological stacks
... This theorem implies that every diagram of topological stacks has a natural weak homotopy type as a diagram of topological spaces. Furthermore, the transformation ϕ relates the given diagram of stacks with its weak homotopy type, thus allowing one to transport homotopical information back and forth ...
... This theorem implies that every diagram of topological stacks has a natural weak homotopy type as a diagram of topological spaces. Furthermore, the transformation ϕ relates the given diagram of stacks with its weak homotopy type, thus allowing one to transport homotopical information back and forth ...
The derived category of sheaves and the Poincare-Verdier duality
... is exact, i.e. f g 0 and ker f Im g . To prove the first part we will show that gf 0 P D pAq. From the normalization axiom we get a distinguished triangle pX, X, 0; 1X , 0, 0q. From the completion axiom we can find 0 Ñ X to complete a commutative diagram ...
... is exact, i.e. f g 0 and ker f Im g . To prove the first part we will show that gf 0 P D pAq. From the normalization axiom we get a distinguished triangle pX, X, 0; 1X , 0, 0q. From the completion axiom we can find 0 Ñ X to complete a commutative diagram ...
subgroups of free topological groups and free
... Schreier transversal exists, and we give such a result in the category of Hausdorff /
... Schreier transversal exists, and we give such a result in the category of Hausdorff /
SIMPLICIAL APPROXIMATION Introduction
... Simplicial approximation theory is a part of the classical literature [1],[2], but it was never developed in a way that was systematic enough to lead to results about model structures. That gap is addressed here: the theory of the subdivision and dual subdivision is developed, both for simplicial co ...
... Simplicial approximation theory is a part of the classical literature [1],[2], but it was never developed in a way that was systematic enough to lead to results about model structures. That gap is addressed here: the theory of the subdivision and dual subdivision is developed, both for simplicial co ...
SIMPLICIAL APPROXIMATION Introduction
... Simplicial approximation theory is a part of the classical literature [1],[2], but it was never developed in a way that was systematic enough to lead to results about model structures. That gap is addressed here: the theory of the subdivision and dual subdivision is developed, both for simplicial co ...
... Simplicial approximation theory is a part of the classical literature [1],[2], but it was never developed in a way that was systematic enough to lead to results about model structures. That gap is addressed here: the theory of the subdivision and dual subdivision is developed, both for simplicial co ...
STRONG HOMOTOPY TYPES, NERVES AND COLLAPSES 1
... theory of finite topological spaces. One can associate to any finite simplicial complex K, a finite T0 -space X (K) which corresponds to the poset of simplices of K. Conversely, one can associate to a given finite T0 -space X the simplicial complex K(X) of its non-empty chains. In [2] we have introd ...
... theory of finite topological spaces. One can associate to any finite simplicial complex K, a finite T0 -space X (K) which corresponds to the poset of simplices of K. Conversely, one can associate to a given finite T0 -space X the simplicial complex K(X) of its non-empty chains. In [2] we have introd ...
Section 3.2 - Cohomology of Sheaves
... the canonical natural equivalence HA,I (X, −) = HA,I 0 (X, −). This means that the A-module structure induced on H i (X, F ) is independent of the choice of resolutions on Mod(X). Definition 5. Let (X, OX ) be a ringed space and A = Γ(X, OX ). Fix an assignment of injective resolutions J to the obje ...
... the canonical natural equivalence HA,I (X, −) = HA,I 0 (X, −). This means that the A-module structure induced on H i (X, F ) is independent of the choice of resolutions on Mod(X). Definition 5. Let (X, OX ) be a ringed space and A = Γ(X, OX ). Fix an assignment of injective resolutions J to the obje ...
Relative Stanley–Reisner theory and Upper Bound Theorems for
... This was the starting point of Stanley–Reisner theory. Stanley’s work spawned extensions of the UBT to (pseudo-)manifolds with (mild) singularities; see for example [Nov03, Nov05, MNS11, NS12]. A pivotal result was a formula of Schenzel [Sch81] that relates algebraic properties of k[] to the face n ...
... This was the starting point of Stanley–Reisner theory. Stanley’s work spawned extensions of the UBT to (pseudo-)manifolds with (mild) singularities; see for example [Nov03, Nov05, MNS11, NS12]. A pivotal result was a formula of Schenzel [Sch81] that relates algebraic properties of k[] to the face n ...
Algebraic K-theory of rings from a topological viewpoint
... paper is to describe some of them. This will give us the opportunity to introduce the definition of the groups Ki (R) for all integers i ≥ 0 (in Sections 1, 2 and 3), to explore their structure (in Section 4) and to present classical results (in Sections 5 and 8). Moreover, the second part of the pa ...
... paper is to describe some of them. This will give us the opportunity to introduce the definition of the groups Ki (R) for all integers i ≥ 0 (in Sections 1, 2 and 3), to explore their structure (in Section 4) and to present classical results (in Sections 5 and 8). Moreover, the second part of the pa ...
A model structure for quasi-categories
... theory” can be regarded as a category with some class of weak equivalences that one would like to formally invert. Any such homotopy theory gives rise to a simplicial category, and conversely simplicial categories arise from homotopy theories up to Dwyer-Kan equivalence. Thus a model structure on th ...
... theory” can be regarded as a category with some class of weak equivalences that one would like to formally invert. Any such homotopy theory gives rise to a simplicial category, and conversely simplicial categories arise from homotopy theories up to Dwyer-Kan equivalence. Thus a model structure on th ...
Sheaves of Groups and Rings
... Definition 1. Let X be a topological space and U ⊆ X an open subset. An open cover of U is a set S of open subsets of U , with the property that the union set of S is U . Note that the empty set S = ∅ is an open cover of ∅, as is the set S = {∅}. If we want to exclude the trivial case where S is emp ...
... Definition 1. Let X be a topological space and U ⊆ X an open subset. An open cover of U is a set S of open subsets of U , with the property that the union set of S is U . Note that the empty set S = ∅ is an open cover of ∅, as is the set S = {∅}. If we want to exclude the trivial case where S is emp ...
A Topology Primer
... point I rather wish to make is that continuity of mappings can be phrased in terms of openness of sets, and as we will see shortly a set of very simple axioms governing this notion is all that is needed to do so. Thus we have arrived at the most basic of all definitions in topology. 1.4 Definition L ...
... point I rather wish to make is that continuity of mappings can be phrased in terms of openness of sets, and as we will see shortly a set of very simple axioms governing this notion is all that is needed to do so. Thus we have arrived at the most basic of all definitions in topology. 1.4 Definition L ...
Spectra for commutative algebraists.
... K∗ (R) = π∗ (K(R)). Examples from geometric topology include the Whitehead space W h(X), Waldhausen’s K-theory of spaces A(X) [33] and the classifying space of the stable mapping class group BΓ+ ∞ [32]. We will give further details of some of these constructions later. 2.D. Fourth answer. This, fina ...
... K∗ (R) = π∗ (K(R)). Examples from geometric topology include the Whitehead space W h(X), Waldhausen’s K-theory of spaces A(X) [33] and the classifying space of the stable mapping class group BΓ+ ∞ [32]. We will give further details of some of these constructions later. 2.D. Fourth answer. This, fina ...
Spectra for commutative algebraists.
... K∗ (R) = π∗ (K(R)). Examples from geometric topology include the Whitehead space W h(X), Waldhausen’s K-theory of spaces A(X) [33] and the classifying space of the stable mapping class group BΓ+ ∞ [32]. We will give further details of some of these constructions later. 2.D. Fourth answer. This, fina ...
... K∗ (R) = π∗ (K(R)). Examples from geometric topology include the Whitehead space W h(X), Waldhausen’s K-theory of spaces A(X) [33] and the classifying space of the stable mapping class group BΓ+ ∞ [32]. We will give further details of some of these constructions later. 2.D. Fourth answer. This, fina ...
Sheaf Cohomology 1. Computing by acyclic resolutions
... below which will have the property δθ + θδ = 1 on C q (U, S)x from which it will follow that the higher joints are exact. (This θ is a fragment of a chain homotopy). Remarks: Varying the choice of U does affect θ. We construct θ as follows, depending upon choice of U . For fx ∈ C i (U, S)x , choose ...
... below which will have the property δθ + θδ = 1 on C q (U, S)x from which it will follow that the higher joints are exact. (This θ is a fragment of a chain homotopy). Remarks: Varying the choice of U does affect θ. We construct θ as follows, depending upon choice of U . For fx ∈ C i (U, S)x , choose ...
Fuglede
... proof or reference. Theorem 1 asserts that every open subset of a polyhedron is a polyhedron. This is crucial for the theory of locally defined harmonic maps with polyhedral domain. A quite brief indication of proof was given by Spanier [Sp, p. 149] for the case of a compact polyhedron. The proof us ...
... proof or reference. Theorem 1 asserts that every open subset of a polyhedron is a polyhedron. This is crucial for the theory of locally defined harmonic maps with polyhedral domain. A quite brief indication of proof was given by Spanier [Sp, p. 149] for the case of a compact polyhedron. The proof us ...
Homology (mathematics)
In mathematics (especially algebraic topology and abstract algebra), homology (in part from Greek ὁμός homos ""identical"") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. See singular homology for a concrete version for topological spaces, or group cohomology for a concrete version for groups.For a topological space, the homology groups are generally much easier to compute than the homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.The original motivation for defining homology groups is the observation that shapes are distinguished by their holes. But because a hole is ""not there"", it is not immediately obvious how to define a hole, or how to distinguish between different kinds of holes. Homology is a rigorous mathematical method for defining and categorizing holes in a shape. As it turns out, subtle kinds of holes exist that homology cannot ""see"" — in which case homotopy groups may be what is needed.