0708-1300/Class notes for Tuesday, January 8 - Revision history

Trefor at 19:00, 8 February 2008

2008-02-08T19:00:04Z

← Older revision		Revision as of 15:00, 8 February 2008
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	Identity: [e] = [x<sub>0</sub>], where x<sub>0</sub> is the constant path.		Identity: [e] = [x<sub>0</sub>], where x<sub>0</sub> is the constant path.

	Inverse: [γ]<sup>-1</sup> = [<math>\tilde{\gamma}</math>], where <math>\tilde{\gamma}</math>(t) = γ(t).		Inverse: [γ]<sup>-1</sup> = [<math>\tilde{\gamma}</math>], where <math>\tilde{\gamma}</math>(t) = γ(1-t).

	'''BC4:''' π<sub>1</sub> is a functor.		'''BC4:''' π<sub>1</sub> is a functor.
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	3) It respects the identity.		3) It respects the identity.

	4) <math> (gf)_* </math> = <math> f_* ~~</math> • <math>~~ g_* </math>.		4) <math> (f\circ g)_* </math> = <math> f_\circ g_ </math>.

	'''Examples:'''		'''Examples:'''

Trefor at 18:30, 8 February 2008

2008-02-08T18:30:15Z

← Older revision		Revision as of 14:30, 8 February 2008
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	Identity: [e] = [x<sub>0</sub>], where x<sub>0</sub> is the constant path.		Identity: [e] = [x<sub>0</sub>], where x<sub>0</sub> is the constant path.

	Inverse: [γ]<sup>-1</sup> = [&gamma~~; "tilda"~~], where &gamma~~;"tilda"~~(t) = γ(t).		Inverse: [γ]<sup>-1</sup> = [<math>\tilde{\gamma}</math>], where <math>\tilde{\gamma}</math>(t) = γ(t).

	'''BC4:''' π<sub>1</sub> is a functor.		'''BC4:''' π<sub>1</sub> is a functor.

Megan at 05:16, 9 January 2008

2008-01-09T05:16:18Z

← Older revision		Revision as of 01:16, 9 January 2008
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			{{0708-1300/Navigation}}

	==Algebraic Topology==		==Algebraic Topology==

Megan at 05:14, 9 January 2008

2008-01-09T05:14:33Z

@@ Line 1: / Line 1: @@
-Algebraic Topology
+==Algebraic Topology==
 Temporarily, at least, we will no longer assume that all structures are manifolds and that all functions are smooth.  However, functions will still be assumed to be continuous.
-We will also, temporarily, assume that all spaces are pointed spaces.  That is, a space X will be assumed to have a distinguished point x, whether mentioned explicitely or not.
+We will also, temporarily, assume that all spaces are pointed spaces.  That is, a space X will be assumed to have a distinguished point x, whether mentioned explicitly or not.
 General idea of algebraic topology:  to find "functors" from topology to algebra.
@@ Line 16: / Line 16: @@
 ) Composition laws:
-a) a binary function on hom(C): mor(X<sub>1</sub>, X<sub>2</sub>) x mor (X<sub>2</sub>, X<sub>3</sub> --> mor (X<sub>1</sub>, X<sub>3</sub>)
+a) a binary function on hom(C): mor(X<sub>1</sub>, X<sub>2</sub>) x mor (X<sub>2</sub>, X<sub>3</sub> &rarr; mor (X<sub>1</sub>, X<sub>3</sub>)
 b) Identities (for every X, there is an I<sub>X</sub> in mor(X,X))
@@ Line 25: / Line 25: @@
 b) identities are in fact identities.

Megan at 03:59, 9 January 2008

2008-01-09T03:59:23Z

New page

Algebraic Topology

Temporarily, at least, we will no longer assume that all structures are manifolds and that all functions are smooth. However, functions will still be assumed to be continuous.

We will also, temporarily, assume that all spaces are pointed spaces. That is, a space X will be assumed to have a distinguished point x, whether mentioned explicitely or not.

General idea of algebraic topology: to find "functors" from topology to algebra.

'''Informal definition:'''
A category C consists of

1) a collection Obj(C) of objects

2) a class hom(C) of morphisms between these objects, so that for each X1 and X2 in Obj(C) we get a set mor{X1, X2}

3) Composition laws:

a) a binary function on hom(C): mor(X1, X2) x mor (X2, X3 --> mor (X1, X3)

b) Identities (for every X, there is an IX in mor(X,X))

4) Compatibility laws:

a) associativity of composition

b) identities are in fact identities.