# PlanetPhysics/Homotopy Category

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### Homotopy category, fundamental groups and fundamental groupoids

Let us consider first the category ${\displaystyle Top}$ whose objects are topological spaces ${\displaystyle X}$ with a chosen basepoint ${\displaystyle x\in X}$ and whose morphisms are continuous maps ${\displaystyle X\to Y}$ that associate the basepoint of ${\displaystyle Y}$ to the basepoint of ${\displaystyle X}$. The fundamental group of ${\displaystyle X}$ specifies a functor ${\displaystyle Top\to '''G'''}$, with ${\displaystyle '''G'''}$ being the category of groups and group homomorphisms, which is called the fundamental group functor .

### Homotopy category

Next, when one has a suitably defined relation of homotopy between morphisms, or maps, in a category ${\displaystyle U}$ , one can define the homotopy category ${\displaystyle hU}$ as the category whose objects are the same as the objects of ${\displaystyle U}$ , but with morphisms being defined by the homotopy classes of maps; this is in fact the homotopy category of unbased spaces .

### Fundamental groups

We can further require that homotopies on ${\displaystyle Top}$ map each basepoint to a corresponding basepoint, thus leading to the definition of the homotopy category ${\displaystyle hTop}$ of based spaces . Therefore, the fundamental group is a homotopy invariant functor on ${\displaystyle Top}$ , with the meaning that the latter functor factors through a functor ${\displaystyle hTop\to '''G'''}$. A homotopy equivalence in ${\displaystyle U}$ is an isomorphism in ${\displaystyle hTop}$. Thus, based homotopy equivalence induces an isomorphism of fundamental groups.

### Fundamental groupoid

In the general case when one does not choose a basepoint, a fundamental groupoid ${\displaystyle \Pi _{1}(X)}$ of a topological space ${\displaystyle X}$ needs to be defined as the category whose objects are the base points of ${\displaystyle X}$ and whose morphisms ${\displaystyle x\to y}$ are the equivalence classes of paths from ${\displaystyle x}$ to ${\displaystyle y}$.

• Explicitly, the objects of ${\displaystyle \Pi _{1}(X)}$ are the points of ${\displaystyle X}$ ${\displaystyle \mathrm {Obj} (\Pi _{1}(X))=X\,,}$
• morphisms are homotopy classes of paths "rel endpoints" that is ${\displaystyle \mathrm {Hom} _{\Pi _{1}(x)}(x,y)=\mathrm {Paths} (x,y)/\sim \,,}$ where, ${\displaystyle \sim }$ denotes homotopy rel endpoints, and,
• composition of morphisms is defined via piecing together, or concatenation, of paths.

### Fundamental groupoid functor

Therefore, the set of endomorphisms of an object ${\displaystyle x}$ is precisely the fundamental group ${\displaystyle \pi (X,x)}$. One can thus construct the \htmladdnormallink{groupoid {http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} of homotopy equivalence classes}; this construction can be then carried out by utilizing functors from the category ${\displaystyle Top}$ , or its subcategory ${\displaystyle hU}$, to the \htmladdnormallink{category of groupoids {http://planetphysics.us/encyclopedia/GroupoidCategory4.html} and groupoid homomorphisms}, ${\displaystyle Grpd}$. One such functor which associates to each topological space its fundamental (homotopy) groupoid is appropriately called the fundamental groupoid functor.

### An example: the category of simplicial, or CW-complexes

As an important example, one may wish to consider the category of simplicial, or ${\displaystyle CW}$-complexes and homotopy defined for ${\displaystyle CW}$-complexes. Perhaps, the simplest example is that of a one-dimensional ${\displaystyle CW}$-complex, which is a graph. As described above, one can define a functor from the category of graphs, Grph , to ${\displaystyle Grpd}$ and then define the fundamental homotopy groupoids of graphs, hypergraphs, or pseudographs. The case of freely generated graphs (one-dimensional ${\displaystyle CW}$-complexes) is particularly simple and can be computed with a digital computer by a finite algorithm using the finite groupoids associated with such finitely generated ${\displaystyle CW}$-complexes.

#### Remark

Related to this concept of homotopy category for unbased topological spaces, one can then prove the approximation theorem for an arbitrary space by considering a functor ${\displaystyle \Gamma :'''hU'''\longrightarrow '''hU''',}$ and also the construction of an approximation of an arbitrary space ${\displaystyle X}$ as the colimit ${\displaystyle \Gamma X}$ of a sequence of cellular inclusions of ${\displaystyle CW}$-complexes ${\displaystyle X_{1},...,X_{n}}$ , so that one obtains ${\displaystyle X\equiv colim[X_{i}]}$.

Furthermore, the homotopy groups of the ${\displaystyle CW}$-complex ${\displaystyle \Gamma X}$ are the colimits of the homotopy groups of ${\displaystyle X_{n}}$, and ${\displaystyle \gamma _{n+1}:\pi _{q}(X_{n+1})\longmapsto \pi _{q}(X)}$ is a group epimorphism.

## References

1. May, J.P. 1999, A Concise Course in Algebraic Topology. , The University of Chicago Press: Chicago
2. R. Brown and G. Janelidze.(2004). Galois theory and a new homotopy double groupoid of a map of spaces.(2004). Applied Categorical Structures ,12 : 63-80. Pdf file in arxiv: math.AT/0208211