R. Brown, Topology:a geometric account of general topology, homotopy types and the fundamental groupoid , Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, second edition (1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....# 1 (E ) of a locally connected topos E bounded over S . # Partially supported by the Natural Sciences and Engineering Research Council of Canada. Supported by the Australian Research Council. Since a well known device for calculating knot groups in topology is the van Kampen theorems [34, 29, 33, 6], we decided to look for suitable analogues of the van Kampen theorems for toposes. An abstract categorical framework for van Kampen theorems was given by Brown and Janelidze [7] using the notion of extensive category [18] and expressed in terms of coverings. This led us to formulate and prove a ....

Ronald Brown, Topology: A geometric account of general topology, homotopy types and the fundamental groupoid, Ellis Horwood, Chichester, 1988.

....m Theta A . 1 Theta A g Moreover, we define the path relation f v g : A B if and only if there exists ff : Sigma m Theta A B such that ff : f m g. Remark. Ideas similar to the above, though in the context of undirected paths, constitute the basics of homotopy theory (see e.g. [Bro88]) Convention. When the subscript is omitted in Sigma m , m , m and m , it is assumed to be 1. For testing approximation, paths of length one suffice: Proposition 4.3 f v g if and only if f g. Proof: Because f 0 g if and only if f = g, and if ff : f m 1 g : A B then ff ffi (L( m ) Theta ....

R. Brown. Topology: a geometric account of general topology, homotopy types and the fundamental grupoid. Halsted Press, 1988.

....just cited also shows the extent to which groupoids provide a framework for a unified study 2 of operator algebras, foliations, and index theory. In algebraic topology, the fundamental groupoid of a topological space has been exploited by P. Higgins, R. Brown, and others (see Brown s textbook [3], or Higgins book [9] which is a good general introduction to groupoids as well) in situations where the use of a fixed base point as imposed by the usual fundamental group would be too restrictive. Groupoid methods are thus well adapted to disconnected spaces (which arise frequently when ....

Brown, R., Topology: a geometric account of general topology, homotopy types, and the fundamental groupoid, Halsted Press, New York, 1988.

....book just cited also shows the extent to which groupoids provide a framework for a unified study of operator algebras, foliations, and index theory. In algebraic topology, the fundamental groupoid of a topological space has been exploited by P. Higgins, R. Brown, and others (see Brown s textbook [3]) where the use of a fixed base point as imposed by the usual fundamental group would be too restrictive. Groupoid methods are thus well adapted to disconnected spaces (which arise frequently when connected spaces are cut into pieces for study) and to spaces with fixed point free group : ....

Brown, R., Topology: a geometric account of general topology, homotopy types, and the fundamental groupoid, Halsted Press, New York, 1988.

....deformation we may achieve the same effect in 3 d , addressed below, but for now our intuition with Euclidean space will suffice. In topological language, any two paths with common endpoints in such a simply connected space are automatically homotopic, an equivalence relation on paths [Bro88, Whi49, Whi78]. They become nonhomotopic when a hole appears somewhere in the space between them to inhibit their deformation into one another. Homotopy is ordinarily studied for spaces the movements in which form a group under composition, where homotopy is inhibited only by holes. The typical irreversibility ....

R. Brown. Topology: A geometric account of general topology, homotopy types and the fundamental groupoid. Halsted Press, New York, 1988.

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R. Brown, Topology:a geometric account of general topology, homotopy types and the fundamental groupoid , Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, second edition (1988.

....result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author s book on Topology [3]. This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections. Since the book is out of print, and the result seems not well known, we now advertise it here. It is hoped that this account will also allow wider views of these results, for example in topos ....

....at the same point as the other. Let h : I I X G be a homotopy rel end points a b. The method now is not to lift the homotopy h itself, but to lift pieces of a subdivision of h; it is here that the method differs from that used in the theory of covering spaces given in Section 9. 1 of [3]. Subdivide I I, by lines parallel to the axes, into small squares each of which is mapped by h into a strong canonical neighbourhood in X G. This subdivision determines a sequence of homotopies h i : a i 1 a i , i = 1, 2, n, say, where a 0 = a, a n = b. Keep i fixed for the ....

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R. Brown, 1988, Topology: a geometric account of general topology, homotopy types, and the fundamental groupoid, Ellis-Horwood, Chichester.

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R. Brown, Topology : A Geometric Account of General Topology, Homotopy Types and the Fundamental Groupoid, Ellis Horwood, Chichester, 1988.

....; b) over 1 (B 1 B 2 ; b) Here it is assumed that there is a single base point b; say, contained in B 1 B 2 , which is itself assumed path connected. A common proof of this theorem, as well as the groupoid version (which does not need the choice of base point and assumptions of connectivity) [2, 3], uses the definition of the fundamental groupoid in terms of paths. However, it is well known that there is another proof using covering spaces, sometimes called the tautologous proof . Suppose that B is a good space so that the fundamental groupoid 1 (B) classifies the coverings of B, i.e. ....

R. Brown, Topology: a geometric account of general topology, homotopy types, and the fundamental groupoid, Ellis Horwood, Chichester (1988).

....b) over 1 (B 1 B 2 ; b) Here it is assumed that there is a single base point b; say, contained in B 1 B 2 , which is itself assumed path connected. A common proof of this theorem, as well as the groupoid version (which does not need the choice of base point and assumptions of connectivity) [2, 3], uses the definition of the fundamental groupoid in terms of paths. However, it is well known that there is another proof using covering spaces, sometimes called the tautologous proof . Suppose that B is a good space so that the fundamental groupoid 1 (B) classifies the coverings of B, ....

R. Brown, Topology: a geometric account of general topology, homotopy types, and the fundamental groupoid, Ellis Horwood, Chichester (1988).

....) associating to a pregroupoid W a morphism : W M(W ) to a groupoid M(W ) and which is universal for pregroupoid morphisms to a groupoid. First form the free groupoid F (W ) on the graph W , and denote the inclusion W F (W ) by u 7 [u] Let N be the normal subgroupoid (Higgins [13] Brown [4]) of F (W ) generated by the elements [vu] Gamma1 [v] u] for all u; v 2 W such that vu is defined and belongs to W . Then M(W ) is defined to be the quotient groupoid (loc. cit. F (W ) N . The composition W F (W ) M(W ) is written , and is the required universal morphism. In the case W ....

....that each path component of X admits a simply connected covering space. It is standard that if 1 X is the fundamental groupoid of X, topologised as in Brown and Danesh Naruie [5] and x 2 X, then the target map fi: 1 X) x X is the universal covering map of X based at x (see also Brown [4], Chapter 9) Let G be a star Lie groupoid. The groupoid PiG is defined as follows. As a set, PiG is the union of the stars ( 1 G x ) 1x . The object set of PiG is the same as that of G. The function ff: PiG X maps all of ( 1 G x ) 1x to x, while fi: PiG X is on ( 1 G x ) 1x the ....

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Brown, R., Topology: A Geometric Account of General Topology, Homotopy Types and the Fundamental Groupoid, Ellis Horwood, Chichester; Prentice Hall, New York, 1988.

....an adjunction space obtained from the induced covering spaces. If further, ff is a covering map, and X is a CW complex, then f X may be given the structure of a CW complex [27] We also need a special case of the basic facts on the path components and fundamental group of induced covering maps ([7, 8, 27]) Given the following pullback b A F NaN F NaN F NaN F NaN fflffl ff 0 f X F NaN F NaN fflffl ff A F NaN F NaN f X and points a 2 A; x 2 f X such that fa = ff x; in which ff is a universal covering map and X; A; f X are path connected, then there is a sequence 1 ....

.... f X such that fa = ff x; in which ff is a universal covering map and X; A; f X are path connected, then there is a sequence 1 1 ( b A; a; x) 1 (A; a) f 1 (X; fa) 0 ( b A) 1: 5) This sequence is exact in the sense of sequences arising from fibrations of groupoids [7], which involves an operation of the fundamental group 1 (X; fa) on the set 0 ( b A) of path Theory and Applications of Categories, Vol. 1, No. 3 64 components of b A. It follows that the fundamental group of b A is isomorphic to Ker f , and that 0 ( b A) is bijective with the ....

Brown, R., Topology: a geometric account of general topology, homotopy types and the fundamental groupoid, Ellis Horwood, Chichester, (1988).

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