# Difference between revisions of "Space of path components"

(→As a monoid) |
|||

Line 16: | Line 16: | ||

===As a monoid=== | ===As a monoid=== | ||

− | If <math>X</math> is a [[topological group]], [[topological monoid]], or (most generally) a [[H-space]], then <math>\pi_0(X)</math> naturally acquires the structure of a [[monoid]] | + | If <math>X</math> is a [[topological group]], [[topological monoid]], or (most generally) a [[H-space]], then <math>\pi_0(X)</math> naturally acquires the structure of a [[monoid]]. The structure is obtained by quotienting the original monoid (or H-space multiplication) by the equivalence relation of being path-connected. |

In order to perform this quotienting, we need to justify that the relation of being in the same path component is a congruence for the original monoid (or H-space multiplication). This is indeed guaranteed by the continuity of multiplication: if <math>x_1</math> and <math>y_1</math> are in the same path component, and <math>x_2</math> and <math>y_2</math> are in the same path component, then <math>x_1 * x_2</math> and <math>y_1 * y_2</math> are in the same path component, because the paths can be multiplied pointwise. | In order to perform this quotienting, we need to justify that the relation of being in the same path component is a congruence for the original monoid (or H-space multiplication). This is indeed guaranteed by the continuity of multiplication: if <math>x_1</math> and <math>y_1</math> are in the same path component, and <math>x_2</math> and <math>y_2</math> are in the same path component, then <math>x_1 * x_2</math> and <math>y_1 * y_2</math> are in the same path component, because the paths can be multiplied pointwise. | ||

Further, even if the original multiplication was not associative but only associative up to homotopy (making it an H-space), the new multiplication is strictly associative because any homotopy must descend to the identity map at the level of path components. | Further, even if the original multiplication was not associative but only associative up to homotopy (making it an H-space), the new multiplication is strictly associative because any homotopy must descend to the identity map at the level of path components. | ||

+ | |||

+ | '''However, the multiplication need not be continuous on <math>\pi_0(X)</math>.''' Therefore, <math>\pi_0(X)</math> nee not acquire the structure of a topological monoid. | ||

+ | |||

+ | == References == | ||

+ | |||

+ | Some work of J. Brazas or P. Fabel may be relevant. |

## Latest revision as of 13:47, 6 July 2019

## Contents

## Definition

### As a set and pointed set

Suppose is a topological space (we sometimes take a based topological space, but the basepoint turns out to be irrelevant). The **set of path components** of , denoted , is defined as follows:

- As a set, it is the set of path components of . In other words, it is the set of equivalence classes of under the equivalence relation iff there is a path from to in .
- If is nonempty and , then , as a
*pointed set*, is defined as the set with the chosen basepoint being the element of that is the path component of .

### As a topological space

As a topological space, the **space of path components** is obtained by taking the quotient topology of under the equivalence relation iff there is a path from to in .

For a locally path-connected space, the space of path components is a discrete space. Thus, often, when dealing with such spaces, we ignore the topology.

### As a monoid

If is a topological group, topological monoid, or (most generally) a H-space, then naturally acquires the structure of a monoid. The structure is obtained by quotienting the original monoid (or H-space multiplication) by the equivalence relation of being path-connected.

In order to perform this quotienting, we need to justify that the relation of being in the same path component is a congruence for the original monoid (or H-space multiplication). This is indeed guaranteed by the continuity of multiplication: if and are in the same path component, and and are in the same path component, then and are in the same path component, because the paths can be multiplied pointwise.

Further, even if the original multiplication was not associative but only associative up to homotopy (making it an H-space), the new multiplication is strictly associative because any homotopy must descend to the identity map at the level of path components.

**However, the multiplication need not be continuous on .** Therefore, nee not acquire the structure of a topological monoid.

## References

Some work of J. Brazas or P. Fabel may be relevant.