Homotopy category

In mathematics , the homotopy category is the category whose objects are the topological spaces and whose morphisms are the homotopy classes of continuous mappings . It is called hTop .

Explanation

Homotopy defines an equivalence relation on the set of continuous mappings between two topological spaces. The equivalence classes are called homotopy classes . With one notes the totality of the homotopy classes between the topological spaces and . ${\ displaystyle [X, Y]}$ ${\ displaystyle X}$ ${\ displaystyle Y}$

While Top represents the classic category of topological spaces and continuous functions, the morphisms of the category hTop are precisely the homotopy classes. The objects of both categories are the same.

In other words it is

{\ displaystyle \ operatorname {Ob} (\ mathbf {hTop}) = \ operatorname {Ob} (\ mathbf {Top}) = \ {X | X \ {\ text {topological space}} \}}

and for any two objects is considered ${\ displaystyle X, Y \ in \ operatorname {Ob} (\ mathbf {hTop})}$

{\ displaystyle \ operatorname {Mor} (X, Y) = \ left [X, Y \ right]}

.

The morphisms are linked representative-wise, i.e. for topological spaces and continuous mappings, the following applies: ${\ displaystyle X, Y, Z}$ ${\ displaystyle f \ colon X \ to Y, \ g \ colon Y \ to Z}$

{\ displaystyle \ left [g \ right] \ circ \ left [f \ right] = \ left [g \ circ f \ right] \ in \ operatorname {Mor} (X, Z)}

This is well-defined because the homotopy relation is compatible with the execution of functions one after the other .

From this it follows that for a space the identity morphism is always the class of all mappings homotopic to the identical mapping : ${\ displaystyle X}$ ${\ displaystyle Id_ {X} \ colon X \ to X, x \ mapsto x}$

{\ displaystyle \ operatorname {id} _ {X} = \ left [Id_ {X} \ right] \ in \ operatorname {Mor} (X, X)}

properties

The homotopy category is a symmetrical monoidal category with the smash product as the product and the 0 sphere as the neutral element. ${\ displaystyle S ^ {0}}$

The isomorphisms of the homotopy category are the homotopy equivalences of the top category .

The category hTop has its own significance , as it does not consist of sets with an additional structure or functions that are compatible with this structure. Nor can it be construed as such. This means that the homotopy category cannot be made concrete , there is no true functor in the category set of sets.

Generalizations

The homotopy category of (any) model category is obtained by localizing it with respect to the set of weak equivalences.

Individual evidence

^ Edwin H. Spanier: Algebraic Topology. 1st corrected Springer edition. Springer, New York et al. 1981, ISBN 3-540-90646-0 .
^ Peter Freyd : Homotopy is not concrete. In: Franklin P. Peterson (Ed.): The Steenrod Algebra and its Applications. A Conference to Celebrate NE Steenrod's 60th Birthday. Proceedings of the Conference held at the Battelle Memorial Institute, Columbus, Ohio March 30th - April 4th, 1970 (= Lecture Notes in Mathematics . 168). Springer, Berlin et al. 1970, ISBN 3-540-05300-X , pp. 25-34.

[1] Edwin H. Spanier: Algebraic Topology. 1st corrected Springer edition. Springer, New York et al. 1981, ISBN 3-540-90646-0 .

[2] Peter Freyd : Homotopy is not concrete. In: Franklin P. Peterson (Ed.): The Steenrod Algebra and its Applications. A Conference to Celebrate NE Steenrod's 60th Birthday. Proceedings of the Conference held at the Battelle Memorial Institute, Columbus, Ohio March 30th - April 4th, 1970 (= Lecture Notes in Mathematics . 168). Springer, Berlin et al. 1970, ISBN 3-540-05300-X , pp. 25-34.