# 6.6 Cell complexes

In classical topology   , a cell complex is a space obtained by successively attaching discs along their boundaries. It is called a CW complex if the boundary of an $n$-dimensional disc is constrained to lie in the discs of dimension  strictly less than $n$ (the $(n-1)$-skeleton).

Any finite CW complex can be presented as a higher inductive type, by turning $n$-dimensional discs into $n$-dimensional paths and partitioning the image of the attaching map into a source and a target, with each written as a composite of lower dimensional paths. Our explicit definitions of $\mathbb{S}^{1}$ and $\mathbb{S}^{2}$ in §6.4 (http://planetmath.org/64circlesandspheres) had this form.

Another example is the torus $T^{2}$, which is generated by:

• a point $b:T^{2}$,

• a path $p:b=b$,

• another path $q:b=b$, and

• a 2-path $t:p\mathchoice{\mathbin{\raisebox{2.15pt}{\displaystyle\centerdot}}}{% \mathbin{\raisebox{2.15pt}{\centerdot}}}{\mathbin{\raisebox{1.075pt}{% \scriptstyle\,\centerdot\,}}}{\mathbin{\raisebox{0.43pt}{\scriptscriptstyle% \,\centerdot\,}}}q=q\mathchoice{\mathbin{\raisebox{2.15pt}{\displaystyle% \centerdot}}}{\mathbin{\raisebox{2.15pt}{\centerdot}}}{\mathbin{\raisebox{1% .075pt}{\scriptstyle\,\centerdot\,}}}{\mathbin{\raisebox{0.43pt}{% \scriptscriptstyle\,\centerdot\,}}}p$.

Perhaps the easiest way to see that this is a torus is to start with a rectangle, having four corners $a,b,c,d$, four edges $p,q,r,s$, and an interior which is manifestly a 2-path $t$ from $p\mathchoice{\mathbin{\raisebox{2.15pt}{\displaystyle\centerdot}}}{\mathbin{% \raisebox{2.15pt}{\centerdot}}}{\mathbin{\raisebox{1.075pt}{\scriptstyle\,% \centerdot\,}}}{\mathbin{\raisebox{0.43pt}{\scriptscriptstyle\,\centerdot\,% }}}q$ to $r\mathchoice{\mathbin{\raisebox{2.15pt}{\displaystyle\centerdot}}}{\mathbin{% \raisebox{2.15pt}{\centerdot}}}{\mathbin{\raisebox{1.075pt}{\scriptstyle\,% \centerdot\,}}}{\mathbin{\raisebox{0.43pt}{\scriptscriptstyle\,\centerdot\,% }}}s$:

Now identify the edge $r$ with $q$ and the edge $s$ with $p$, resulting in also identifying all four corners. Topologically, this identification can be seen to produce a torus.

The induction  principle for the torus is the trickiest of any we’ve written out so far. Given $P:T^{2}\to\mathcal{U}$, for a section   $\mathchoice{\prod_{x:T^{2}}\,}{\mathchoice{{\textstyle\prod_{(x:T^{2})}}}{% \prod_{(x:T^{2})}}{\prod_{(x:T^{2})}}{\prod_{(x:T^{2})}}}{\mathchoice{{% \textstyle\prod_{(x:T^{2})}}}{\prod_{(x:T^{2})}}{\prod_{(x:T^{2})}}{\prod_{(x:% T^{2})}}}{\mathchoice{{\textstyle\prod_{(x:T^{2})}}}{\prod_{(x:T^{2})}}{\prod_% {(x:T^{2})}}{\prod_{(x:T^{2})}}}P(x)$ we require

• a point $b^{\prime}:P(b)$,

• a path $p^{\prime}:b^{\prime}=^{P}_{p}b^{\prime}$,

• a path $q^{\prime}:b^{\prime}=^{P}_{q}b^{\prime}$, and

• a 2-path $t^{\prime}$ between the “composites” $p^{\prime}\mathchoice{\mathbin{\raisebox{2.15pt}{\displaystyle\centerdot}}}{% \mathbin{\raisebox{2.15pt}{\centerdot}}}{\mathbin{\raisebox{1.075pt}{% \scriptstyle\,\centerdot\,}}}{\mathbin{\raisebox{0.43pt}{\scriptscriptstyle% \,\centerdot\,}}}q^{\prime}$ and $q^{\prime}\mathchoice{\mathbin{\raisebox{2.15pt}{\displaystyle\centerdot}}}{% \mathbin{\raisebox{2.15pt}{\centerdot}}}{\mathbin{\raisebox{1.075pt}{% \scriptstyle\,\centerdot\,}}}{\mathbin{\raisebox{0.43pt}{\scriptscriptstyle% \,\centerdot\,}}}p^{\prime}$, lying over $t$.

In order to make sense of this last datum, we need a composition operation for dependent paths, but this is not hard to define. Then the induction principle gives a function $f:\mathchoice{\prod_{x:T^{2}}\,}{\mathchoice{{\textstyle\prod_{(x:T^{2})}}}{% \prod_{(x:T^{2})}}{\prod_{(x:T^{2})}}{\prod_{(x:T^{2})}}}{\mathchoice{{% \textstyle\prod_{(x:T^{2})}}}{\prod_{(x:T^{2})}}{\prod_{(x:T^{2})}}{\prod_{(x:% T^{2})}}}{\mathchoice{{\textstyle\prod_{(x:T^{2})}}}{\prod_{(x:T^{2})}}{\prod_% {(x:T^{2})}}{\prod_{(x:T^{2})}}}P(x)$ such that $f(b)\equiv b^{\prime}$ and $\mathsf{apd}_{f}\mathopen{}\left(p\right)\mathclose{}=p^{\prime}$ and $\mathsf{apd}_{f}\mathopen{}\left(q\right)\mathclose{}=q^{\prime}$ and something like “$\mathsf{apd}^{2}_{f}\mathopen{}\left(t\right)\mathclose{}=t^{\prime}$”. However, this is not well-typed as it stands, firstly because the equalities $\mathsf{apd}_{f}\mathopen{}\left(p\right)\mathclose{}=p^{\prime}$ and $\mathsf{apd}_{f}\mathopen{}\left(q\right)\mathclose{}=q^{\prime}$ are not judgmental, and secondly because $\mathsf{apd}_{f}$ only preserves path concatenation up to homotopy. We leave the details to the reader (see http://planetmath.org/node/87797Exercise 6.1).

Of course, another definition of the torus is $T^{2}:\!\!\equiv\mathbb{S}^{1}\times\mathbb{S}^{1}$ (in http://planetmath.org/node/87800Exercise 6.3 we ask the reader to verify the equivalence of the two). The cell-complex definition, however, generalizes easily to other spaces without such descriptions, such as the Klein bottle, the projective plane  , etc. But it does get increasingly difficult to write down the induction principles, requiring us to define notions of dependent $n$-paths and of $\mathsf{apd}$ acting on $n$-paths. Fortunately, once we have the spheres in hand, there is a way around this.

Title 6.6 Cell complexes
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