AN INTRODUCTION TO ∞-CATEGORIES Contents 1. Introduction 1
... whether there’s a path between them – that is, whether the two maps are homotopic. This makes the fundamental group (and homotopy groups in general) easy to work with, and gives rise to incredibly useful notions like homotopy equivalence. The notion that the set of morphisms is a space is useful in ...
... whether there’s a path between them – that is, whether the two maps are homotopic. This makes the fundamental group (and homotopy groups in general) easy to work with, and gives rise to incredibly useful notions like homotopy equivalence. The notion that the set of morphisms is a space is useful in ...
Higher algebra and topological quantum field theory
... and target maps. The induced equivalence relation on Mori (x, y) is also denoted ∼. 5. (COMP) for any 0 < i ≤ n and any three i − 1-morphisms u, v and w sharing the same source and target, there is a well-defined composition map (Mori (u, v)/ ∼) × (Mori (v, w)/ ∼) → (Mori (u, w)/ ∼) which is associa ...
... and target maps. The induced equivalence relation on Mori (x, y) is also denoted ∼. 5. (COMP) for any 0 < i ≤ n and any three i − 1-morphisms u, v and w sharing the same source and target, there is a well-defined composition map (Mori (u, v)/ ∼) × (Mori (v, w)/ ∼) → (Mori (u, w)/ ∼) which is associa ...
Hyperbolic Geometry: Isometry Groups of Hyperbolic
... the path, y varies hyperbolically (by a factor of y1 ) from a to b as t increases, and γ is thus a vertical line segment from ia to ib with length ln ab . Since the length h(λ) of any arbitrary path λ between these points is greater than or equal to ln ab , the shortest possible path must be the str ...
... the path, y varies hyperbolically (by a factor of y1 ) from a to b as t increases, and γ is thus a vertical line segment from ia to ib with length ln ab . Since the length h(λ) of any arbitrary path λ between these points is greater than or equal to ln ab , the shortest possible path must be the str ...
Locally compact groups and continuous logic
... continuous logic can work in these cases. In fact metric groups from these classes can be presented as reducts of continuous metric structures which induce required actions. In the second part of the section we consider non-(T) and non-FR (of fixed points for isometric actions on real trees). Note t ...
... continuous logic can work in these cases. In fact metric groups from these classes can be presented as reducts of continuous metric structures which induce required actions. In the second part of the section we consider non-(T) and non-FR (of fixed points for isometric actions on real trees). Note t ...
NONPOSITIVE CURVATURE AND REFLECTION GROUPS Michael
... I. Nonpositively Curved Spaces. The notion of “nonpositive curvature” (or more generally of “curvature bounded above by a real number ” makes sense for a more general class of metric spaces than Riemannian manifolds: one need only assume that any two points can be connected by a geodesic segment. F ...
... I. Nonpositively Curved Spaces. The notion of “nonpositive curvature” (or more generally of “curvature bounded above by a real number ” makes sense for a more general class of metric spaces than Riemannian manifolds: one need only assume that any two points can be connected by a geodesic segment. F ...
Proper actions on topological groups: Applications to quotient spaces
... In 1961 Palais [23] introduced the very important concept of a proper action of an arbitrary locally compact group G and extended a substantial part of the theory of compact Lie transformation groups to noncompact ones. Let X be a G-space. Two subsets U and V in X are called thin relative to each ot ...
... In 1961 Palais [23] introduced the very important concept of a proper action of an arbitrary locally compact group G and extended a substantial part of the theory of compact Lie transformation groups to noncompact ones. Let X be a G-space. Two subsets U and V in X are called thin relative to each ot ...
Topological groups: local versus global
... Indeed, the following general statement holds [2]: Lemma 2.1. Let P be a topological property preserved by preimages of spaces under perfect mappings (in the class of Tychonoff spaces) and also inherited by closed sets. Suppose further that G is a topological group and H is a locally compact subgrou ...
... Indeed, the following general statement holds [2]: Lemma 2.1. Let P be a topological property preserved by preimages of spaces under perfect mappings (in the class of Tychonoff spaces) and also inherited by closed sets. Suppose further that G is a topological group and H is a locally compact subgrou ...
Lectures on quasi-isometric rigidity
... Groups ←→ Metric Spaces The first step is to associate with a finitely-generated group G a metric space X. Let G be a group with a finite generating set S = {s1 , ..., sk }. Then we construct a graph X, whose vertex set V (X) is the group G itself and whose edges are [g, gsi ], si ∈ S, g ∈ G. (If gs ...
... Groups ←→ Metric Spaces The first step is to associate with a finitely-generated group G a metric space X. Let G be a group with a finite generating set S = {s1 , ..., sk }. Then we construct a graph X, whose vertex set V (X) is the group G itself and whose edges are [g, gsi ], si ∈ S, g ∈ G. (If gs ...
THE REAL DEFINITION OF A SMOOTH MANIFOLD 1. Topological
... 5. The Hausdorff property and manifolds In this section, we drop temporarily the assumption that manifolds are Hausdorff in order to discuss a useful property which is equivalent to the Hausdorff property for manifolds. Proposition 4. Let X be a manifold of dimension n. The following property is equ ...
... 5. The Hausdorff property and manifolds In this section, we drop temporarily the assumption that manifolds are Hausdorff in order to discuss a useful property which is equivalent to the Hausdorff property for manifolds. Proposition 4. Let X be a manifold of dimension n. The following property is equ ...
fixed points and admissible sets
... general convex structure G(x, y, λ) satisfying conditions (1) and (2). Let F = {T | T : X → X} be a finite commuting family of nonexpansive mappings of X into X than F has a common fixed point. Proof. Let Φ be a family of all nonempty, admissible subsets of X, each of which is mapped into itself by f ...
... general convex structure G(x, y, λ) satisfying conditions (1) and (2). Let F = {T | T : X → X} be a finite commuting family of nonexpansive mappings of X into X than F has a common fixed point. Proof. Let Φ be a family of all nonempty, admissible subsets of X, each of which is mapped into itself by f ...
spaces every quotient of which is metrizable
... [1, 2, 6, 7]. In an elementary course in general topology we learn that every continuous image in Hausdorff space of a compact metric space is metrizable [8]. In [7], Willard proved that every closed continuous image of a metric space X in Hausdorff space is metrizable if and only if the set of all ...
... [1, 2, 6, 7]. In an elementary course in general topology we learn that every continuous image in Hausdorff space of a compact metric space is metrizable [8]. In [7], Willard proved that every closed continuous image of a metric space X in Hausdorff space is metrizable if and only if the set of all ...
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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 ...
Manifolds
... A family At , t ∈ I of subsets of a topological space X is called an isotopy of the set A = A0 , if the graph Γ = {(x, t) ∈ X × I | x ∈ At } of the family is fibrewise homeomorphic to the cylinder A × I, i. e. there exists a homeomorphism A × I → Γ mapping A × {t} to Γ ∩ X × {t} for any t ∈ I. Such ...
... A family At , t ∈ I of subsets of a topological space X is called an isotopy of the set A = A0 , if the graph Γ = {(x, t) ∈ X × I | x ∈ At } of the family is fibrewise homeomorphic to the cylinder A × I, i. e. there exists a homeomorphism A × I → Γ mapping A × {t} to Γ ∩ X × {t} for any t ∈ I. Such ...
Finite topological spaces
... If (X , x0 ) and (Y , y0 ) are ponited spaces and f and g satisfy f (x0 ) = g (x0 ) = y0 , then we also require that F (x0 , t) = y0 for every t. Homotopy equivalence of maps is an equivalence relation. ...
... If (X , x0 ) and (Y , y0 ) are ponited spaces and f and g satisfy f (x0 ) = g (x0 ) = y0 , then we also require that F (x0 , t) = y0 for every t. Homotopy equivalence of maps is an equivalence relation. ...
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... the origin. This triangulation is invariant by the 2π/n rotation. Now let C be the complex obtained by attaching the central vertex of D 2 to some vertex of BΓ . It is clear that Aut(C) contains Z/nZ. Since n ≥ 2 was arbitrary, the conclusions both of Theorem 1.1 and of Theorem 1.2 do not hold. 3. T ...
... the origin. This triangulation is invariant by the 2π/n rotation. Now let C be the complex obtained by attaching the central vertex of D 2 to some vertex of BΓ . It is clear that Aut(C) contains Z/nZ. Since n ≥ 2 was arbitrary, the conclusions both of Theorem 1.1 and of Theorem 1.2 do not hold. 3. T ...
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... begin with two disjoint subsets C; D R and for each x 2 D a sequence hxn i in C converging to x. They let X(C; D) be the union C [ D but with points of C isolated and neighbourhoods of points of D containing tails of the corresponding sequences. The essential features of X(C; D) are then preserved ...
... begin with two disjoint subsets C; D R and for each x 2 D a sequence hxn i in C converging to x. They let X(C; D) be the union C [ D but with points of C isolated and neighbourhoods of points of D containing tails of the corresponding sequences. The essential features of X(C; D) are then preserved ...
RIGID RATIONAL HOMOTOPY THEORY AND
... (2) Use de Jong’s alterations and cohomological descent to show that for any X{k we can choose some simplicial k-scheme X‚ Ñ X, with each Xn of the above form, such that RΓrig pX{Kq » holim∆ RΓrig pX‚ {Kq as Frobenius cdga’s. (3) Show that if A˚,‚ is a cosimplicial cdga, each with Frobenius structur ...
... (2) Use de Jong’s alterations and cohomological descent to show that for any X{k we can choose some simplicial k-scheme X‚ Ñ X, with each Xn of the above form, such that RΓrig pX{Kq » holim∆ RΓrig pX‚ {Kq as Frobenius cdga’s. (3) Show that if A˚,‚ is a cosimplicial cdga, each with Frobenius structur ...
THE GEOMETRIES OF 3
... division of closed Seifert fibre spaces into six types corresponding to six of the geometries. Finally, in §6, I discuss briefly the progress made so far in proving Thurston's Geometrisation Conjecture. None of the results of this article are really new, but the treatment of some of the topics is ne ...
... division of closed Seifert fibre spaces into six types corresponding to six of the geometries. Finally, in §6, I discuss briefly the progress made so far in proving Thurston's Geometrisation Conjecture. None of the results of this article are really new, but the treatment of some of the topics is ne ...
BASIC TOPOLOGICAL FACTS 1. вга дб 2. егждб § ¥ ¨ ждб 3. ейдб
... is an open map. Important Point: Condition 3 does NOT imply that Thus we can NOT say that an open subset of is an open subset of . The problem is that for any generic subset , is not necessarily saturated. In general, quotient maps need not be injective. Proposition: ...
... is an open map. Important Point: Condition 3 does NOT imply that Thus we can NOT say that an open subset of is an open subset of . The problem is that for any generic subset , is not necessarily saturated. In general, quotient maps need not be injective. Proposition: ...
SimpCxes.pdf
... We could remove the word finite from the previous definition by defining geometric simplicial complexes more generally, without reference to a finite dimensional ambient space RN , but there is no point in going into that here. We also note that we do not have to realize in such a high dimensional E ...
... We could remove the word finite from the previous definition by defining geometric simplicial complexes more generally, without reference to a finite dimensional ambient space RN , but there is no point in going into that here. We also note that we do not have to realize in such a high dimensional E ...
FINITE SPACES AND SIMPLICIAL COMPLEXES 1. Statements of
... We could remove the word finite from the previous definition by defining geometric simplicial complexes more generally, without reference to a finite dimensional ambient space RN , but there is no point in going into that here. We also note that we do not have to realize in such a high dimensional E ...
... We could remove the word finite from the previous definition by defining geometric simplicial complexes more generally, without reference to a finite dimensional ambient space RN , but there is no point in going into that here. We also note that we do not have to realize in such a high dimensional E ...
On Colimits in Various Categories of Manifolds
... proves U cannot be homeomorphic to Rn for any n. This example actually shows even more than we expected, because the group which is acting is a Lie group. So quotients of manifolds by Lie groups need not be manifolds. The standard way to fix this is to place more restrictions on the type of action. ...
... proves U cannot be homeomorphic to Rn for any n. This example actually shows even more than we expected, because the group which is acting is a Lie group. So quotients of manifolds by Lie groups need not be manifolds. The standard way to fix this is to place more restrictions on the type of action. ...
FINITE SPACES AND SIMPLICIAL COMPLEXES 1. Statements of
... well-defined and continuous. If g is a bijection on vertices and simplices, we say that it is an isomorphism, and then |g| is a homeomorphism. It is usual to abbreviate |g| to g and to refer to it as a simplicial map. Definition 4.6. The abstract simplicial complex aK determined by a geometric simpl ...
... well-defined and continuous. If g is a bijection on vertices and simplices, we say that it is an isomorphism, and then |g| is a homeomorphism. It is usual to abbreviate |g| to g and to refer to it as a simplicial map. Definition 4.6. The abstract simplicial complex aK determined by a geometric simpl ...
LECTURE NOTES 4: CECH COHOMOLOGY 1
... X 7→ Z(X) = C(X, Z) is a contravariant functor from C to (sets). It is the functor corepresented by Z. Example 1.4. Let A be an abelian group, considered as a space with the discrete topology. Then A(U ) = map(U, A) is a group: if f, g : U → A, then their sum is ...
... X 7→ Z(X) = C(X, Z) is a contravariant functor from C to (sets). It is the functor corepresented by Z. Example 1.4. Let A be an abelian group, considered as a space with the discrete topology. Then A(U ) = map(U, A) is a group: if f, g : U → A, then their sum is ...
HOMEOMORPHISMS THE GROUPS OF AND
... (c) Since (X,T) is not indiscrete, there exists a proper nonempty open set A of (X,T). Let b X\A. Then X\{b} is open by (b). Thus } Is closed. Then every sngleton subset of (X,T) is closed, since (X,T) Is homogeneous by (a). Hence every fnlte subset, being a finite union of sngleton subsets is close ...
... (c) Since (X,T) is not indiscrete, there exists a proper nonempty open set A of (X,T). Let b X\A. Then X\{b} is open by (b). Thus } Is closed. Then every sngleton subset of (X,T) is closed, since (X,T) Is homogeneous by (a). Hence every fnlte subset, being a finite union of sngleton subsets is close ...