Ghost maps

 Seminar topic

Definition: Let $X$ be a topological space based at $x\in X$. Let $PX$ be the space of based paths of $X$, that is, maps $[0,1]\to X$ with $0\mapsto x$. Let $\Omega X\subset PX$ be the space of based loops of $X$, that is, maps $[0,1]\to X$ with $0,1\mapsto x$.

Note that $\Omega$ is a functor on the category of based topoloigcal spaces right-adjoint to the suspension functor $\Sigma$. Also observe there is a fibration
\[
\Omega X \to PX \tov{p} X,
\]
where $p$ is evaluation at $1\in [0,1]$. Since $PX$ is contractible, $H_n(PX)=0$ for $n\neq 0$, so $H_1(\Omega X) \cong H_2(X)$.

Definition: A spectrum $E$ is a sequence of based topological spaces $(E_n,x_n)$ and based homeomorphisms $\alpha_n:E_n \to \Omega E_{n+1}$. A map of spectra $f:E\to F$ is a sequence of based homeomorphisms $f_n:E_n\to F_n$ compatible with the based homeomorphisms of $E$ and $F$, that is, so that the diagram

commutes for all $n$.

Definition: Let $E,F$ be spectra. A map of spectra $f:E\to F$ is a ghost map if the induced map $\pi_nf:\pi_nX \to \pi_n Y$ on stable homotopy groups is the zero map.

Most commonly this term is used in spectra, but the idea of a ghost map may be generalized to other situations, where a map induces the zero map on homology, cohomology, or some similar functor.

References: Weibel (An introduction to homological algebra, Chapters 5.3, 10.9)

The Hodge decomposition, diamond, and Euler characteristics

 Seminar topic

Recall the sheaf of $r$-differential forms $\Omega^r_X$ on $X$ (with $\Omega^r_X(U) = \{fdx_{i_1}\wedge \cdots \wedge dx_{i_r}\ :\ f\ $is well-defined on $U\}$ and such sums) and the structure sheaf $\mathcal O_X$ on $X$ (with $\mathcal O_X(U) = \{f/g\ :\ f,g\in k[U],\ g\neq 0\ $on$\ U\}$). Then we may consider the sheaf cohomology of $X$, with values in $\Omega^r_X$ or $\mathcal O_X$.

Definition: Let $X$ be a smooth manifold of dimension $n$. The $(p,q)$th Hodge number is $h^{p,q}=\dim(H^{p,q})$, where $H^{p,q} = H^q(X,\Omega^p_X)$. These numbers are arranged in a Hodge diamond as below.

The Hodge diamond has a lot of repetition - by complex conjugation, we get that $h^{p,q}=h^{q,p}$, so it is symmetric about its vertical axis. By the Hard Lefschetz theorem (or the Hodge star operator, or Poincare duality), we get that $h^{p,q}=h^{n-q,n-p}$, so it is symmetric about its horizontal axis.

Proposition: Let $X$ be a Kähler manifold (note that all smooth projective varieties are Kähler) of dimension $n$. Then the cohomology groups of $X$ decompose as
\[
H^k(X,\C) = \bigoplus_{p+q=k}H^{p,q}(X),
\]
for all $0\leqslant k\leqslant 2n$. This is called the Hodge decomposition of $X$.

This decomposition immediately gives all the Hodge numbers for $\P^n$, knowing its cohomology. For a manifold of complex dimension $n$, there are several numbers and polynomials that may be defined. These are:
\begin{align*}
\chi_{top}(X) & = \sum_{i=1}^{2n}(-1)^i \dim(H^i(X,\C)) & \text{the (topological) Euler characteristic} \\
\chi^p(X) & = \sum_{q=0}^{n-1}(-1)^qh^{p,q} & \text{the chi-$p$ characteristic} \\
\chi_y(X) & = \sum_{p=0}^{n-1}\chi^py^p & \text{the chi-$y$ characteristic}
\end{align*}
Note the Euler characteristic is the alternating sum of the rows of the Hodge diamond, and the chi-$p$ characteristic is the alternating sum of the left-right diagonals of the diamond.

Example: In the case $X$ is a hypersurface in projective $n$-space $\P^n$ defined by a degree $d$ polynomial,
\[
\chi_y = [z^n]\frac{1}{(1+zy)(1-z)^2}\cdot\frac{(1+zy)^d-(1-z)^d}{(1+zy)^d+y(1-z)^d}.
\]
Since every row except the middle row of the Hodge diamond of a hypersurface is known (as it comes from the Hodge diamond of $\P^n$ by the Lefschetz hyperplane theorem), this expression gives all the unknown numbers. This particular formula is a simplification  of Theorem 22.1.1 in Hirzebruch, which itself comes from the Riemann--Roch theorem.

References: Huybrechts (Complex Geometry: An Introduction, Chapters 3.2, 3.3), Hirzebruch (Topological Methods in Algebraic Geometry, Appendix 1, Section 22)

Exactness and derived functors

 Lecture topic

Let $0\to X\to Y\to Z\to 0$ be a short exact sequence of objects in a category $A$. Let $\mathcal F:A\to B$ be a covariant functor.

Definition: The functor $\mathcal F$ is right-exact if $\mathcal F(X)\to\mathcal F(Y)\to \mathcal F(Z)\to 0$ is an exact sequence. It is left-exact if $0\to \mathcal F(X)\to\mathcal F(Y)\to \mathcal F(Z)$ is an exact sequence. It is exact if it is both left- and right-exact.

Example: These are some examples of left- and right-exact functors:
    $\Hom_A(X,-)$ is covariant left-exact
    $\Hom_A(-,X)$ is contravariant left-exact
    $-\otimes_R X$ is covariant right-exact, for $X$ a left $R$-module

Recall that $X\otimes_R Y$ is naturally isomorphic to $Y\otimes_RX$.

Definition: An object $X\in \Obj(A)$ is projective if $\Hom_A(X,-)$ is an exact functor. Similarly, $X$ is injective if $\Hom_A(-,X)$ is an exact functor.

Recall that a projective resolution of an object $X$ is a sequence of projective objects $\cdots\to P_2\to P_1\to P_0$ that may or may not terminate on the left. The homology of the sequence in degree 0 is $X$, and trivial in other degrees. Similarly, an injective resolution of $X$ is a sequence of injective objects $I_0\to I_1\to I_2\to\cdots$ that may or may not terminate on the right. The cohomology is also concentrated in degree 0, and is $X$ there. A free resolution is a projective resolution where all the objects are free (whatever that means in the context).

These types of resolutions may not exist. A category "has enough injectives (projectives)" means we can always construct injective (projective) resolutions.

Definition: Let $\mathcal F:A\to B$ be a covariant right-exact functor and $\mathcal G:A\to B$ a covariant left-exact functor. Let $X\in \Obj(A)$ with $P_\bullet$ a projective resolution of $X$ and $I_\bullet$ an injective resolution of $X$. The $i$th left-derived functor of $\mathcal F$ is $L_i\mathcal F(X) = H_i(\mathcal F(P_\bullet))$. The $i$th right-derived functor of $\mathcal G$ is $R^i\mathcal G(X) = H^i(\mathcal G(I_\bullet))$.

These objects of $B$ are well-defined up to natural isomorphism. Note that $\mathcal F^{op}:A^{op}\to B^{op}$ is a contravariant right-exact functor. Moreover, if $\mathcal F$ was contravariant right-exact and $\mathcal G$ was contravariant left-exact, then $L_i\mathcal F(X)=H_i(\mathcal F(I_\bullet))$ and $R^i\mathcal G(X)=H^i(\mathcal G(P_\bullet))$.

Example: Let $R$ be a ring with $X$ and $Y$ both $R$-bimodules. Then
\begin{align*}
\Tor_i^R(Y,X) & =  L_i(-\otimes_RX)(Y) &
\Ext^i_R(X,Y) & = R^i(\Hom_R(X,-))(Y) \\
& = L_i(Y\otimes_R - )(X),
&& = R^i(\Hom_R(-,Y))(X).
\end{align*}
Recall that $\Tor_i^R(Y,X)$ is canonically isomorphic to $\Tor_i^R(X,Y)$, but it is not true for $\Ext$. Also note that $\Hom_R(X,-)$ is covariant and $\Hom_R(-,Y)$ is contravariant, while $-\otimes_R X$ and $Y\otimes_R -$ are both covariant functors.
References: Weibel (An introduction to homological algebra, Chapter 2)

Examples of limits and colimits

 Lecture topic

Let $C$ be a category and $X,Y,Z\in \Obj(C)$. Choose $I$ to be a category with $\mathcal F:I\to C$ a functor as described below. Then we may consider the limit and colimit of $\mathcal F$, noting that they may not always exist, as there may be no suitable natural transformation $i$ or $\pi$.

The limit and colimit of the category $I$ with two points and two arrows going between the points in opposite directions, namely

are not interesting to consider. That is because as a category, it must satisfy compositions, so $f\circ g=\id$, which is a restrictive condition on $f$ and $g$. We may define a new map $h:X\to X$ with $h=f\circ g$, but then more maps, such as $h\circ f$ and so on need to be defined, which complicate the situation.

References: Borceux (Handbook of Categorical Algebra I, Chapter 2)

Limits and colimits

 Lecture topic

Definition: Given categories $A,B$ and functors $\mathcal F,\mathcal G:A\to B$, a natural transformation $\eta:\mathcal F\to \mathcal G$ is a collection of elements $\eta_X\in \Hom_B(\mathcal F(X),\mathcal G(X))$ for all $X\in \Obj(A)$ such that the diagram

commutes, whenever $f\in \Hom_A(X,Y)$.

Definition: For $X\in \Obj(A)$, define the constant category $\underline X$ to be the category with $\Obj(\underline X)=\{X\}$ and $\Hom_{\underline X}(X,X)=\{\id_X\}$. For any other category $B$, this may also be viewed as a natural transformation $\underline X:B\to A$ with $\underline X(Y)=X$ and $\underline X(f)=\id_X$ for any object $Y$ and any morphism $f$ of $B$.

Definition: Let $A$ be a small category and $\mathcal F:A\to B$ a functor. The colimit $\text{colim}(\mathcal F$) of $\mathcal F$ is an object $\text{colim}(\mathcal F)\in \Obj(B)$ and a natural transformation $\iota:\mathcal F\to \underline{\text{colim}(\mathcal F)}$ that is initial among all such natural transformations. We write $\iota_X:\mathcal F(X)\to \text{colim}(\mathcal F)$ and have $\iota(f)=\id_{\text{colim}(\mathcal F)}$ for any morphism $f$ of $A$.

In other words, whenever $Z\in \Obj(B)$ and $\eta:\mathcal F\to \underline{Z}$ is a natural transformation, there is a unique map $\zeta:\text{colim}(\mathcal F)\to Z$ such that the following diagram commutes:

Definition: Let $A$ be a small category and $\mathcal F:A\to B$ a functor. The limit $\lim(\mathcal F$) of $\mathcal F$ is an object $\lim(\mathcal F)\in \Obj(B)$ and a natural transformation $\pi:\underline{\lim(\mathcal F)}\to \mathcal F$ that is final among all such natural transformations. We write $\pi_X:\lim(\mathcal F) \to \mathcal F(X)$ and have $\pi(f)=\id_{\lim(\mathcal F)}$ for any morphism $f$ of $A$.

In other words, whenever $Z\in \Obj(B)$ and $\epsilon:\underline{Z}\to \mathcal F$ is a natural transformation, there is a unique map $\theta:Z\to \lim(\mathcal F)$ such that the following diagram commutes:

Examples of colimits are initial objects, coproducts, cokernels, pushouts, direct limits. Examples of limits are final objects, products, kernels, pullbacks, inverse limits.

 Remark: $\Hom$ commutes with limits and tensor commutes with colimits. That is:
\[
\Hom(A,\lim(B_i)) = \lim\left(\Hom(A,B_i)\right)
\hspace{1cm}
(\text{colim}(A_i))\otimes B = \text{colim}(A_i\otimes B)
\]
References: May (A Concise course in Algebraic Topology, Chapter 2.6), Aluffi (Algebra: Chapter 0, Chapter VIII.1)

The canonical bundle of projective space and hypersurfaces

Let $\P^n$ be projective $n$-space with coordinates $[x_0:\cdots:x_n]$. Cover $\P^n$ with affine pieces $U_i = \{x_i\neq 0\}$, each of which are $\A^n$, in coordinates $(y_1,\dots,y_n)$, where $y_j = x_j/x_i$. Recall that the canonical bundle of $\P^n$ is the $n$-fold wedge of the cotangent bundle of $\P^n$, or $\omega_{\P^n} = \bigwedge^nT^*_{\P^n}$. The canonical bundle for an arbitrary variety is defined analogously.

Definition: Let $X$ be a projective $n$-dimensional variety. The sheaf of regular functions on $X$ is $\mathcal O_X$, with $\mathcal O_X(U)=\{f/g\ :\ f,g\in k[x_1,\dots,x_n]/I(X), g\neq 0\}$, and the restriction maps are function restriction.

There is a natural grading on $\mathcal O_X$, given by $\deg(f)-\deg(g)$. A shift in the grading may be applied, called a {\it Serre twist}, to get a differently graded (but isomorphic) module: for $\varphi\in \mathcal O_X$ with $\deg(\varphi)=k$, set $\varphi\in\mathcal O_X(\ell)$ to have $\deg(\varphi) = k-\ell$.

Let $\alpha = dy_1\wedge\cdots\wedge dy_n\in \omega_{\P^n}$, which is well-defined on all of $U_i$. We claim this is well-defined on all of $\P^n$. We check this on the overlap $U_0\cap U_n$ (for nicer notation), but the approach is analogous for $U_i\cap U_j$.
\begin{align*}
U_0 & = \{(y_1,\dots,y_n)\ :\ y_i = x_i/x_0\} & y_i & = \frac{z_{i+1}}{z_i} & dy_i & = \frac{z_1dz_{i+1}-z_{i+1}dz_1}{z_1^2} \\
U_n & = \{(z_1,\dots,z_n)\ :\ z_i = x_{i-1}/x_n\} & y_n & = \frac1{z_1} & dy_n & = \frac{-dz_1}{z_1^2}
\end{align*}
Therefore
\begin{align*}
\alpha & = dy_1\wedge\cdots\wedge dy_n \\
& = \frac{z_1dz_2-z_2dz_1}{z_1^2}\wedge\cdots\wedge \frac{z_1dz_n-z_ndz_1}{z_1^2}\wedge \frac{-dz_1}{z_1^2} \\
& = \frac{dz_2}{z_1}\wedge\cdots\wedge \frac{dz_n}{z_1}\wedge \frac{-dz_1}{z_1^2} \\
& = \frac{(-1)^n}{z_1^{n+1}}dz_1\wedge\cdots \wedge dz_n.
\end{align*}
Since the transition function has a pole of order $n+1$ when $z_1 = 0$, which happens when $x_0=0$, we have that $\alpha$ has a pole of order $n+1$ at $\infty$. Therefore $\omega_{\P^n} \cong \mathcal O_{\P^n}(-n-1)$.

Let $X\subset \P^n$ be a smooth hypersurface defined by a degree $d$ equation $F(x_0,\dots,x_n)=0$. On the affine piece $U_0$ this becomes $f(y_1,\dots,y_n)=F(1,\frac{x_1}{x_0},\dots,\frac{x_n}{x_0})$ with $y_i = x_i/x_0$. The total derivative is
\[
\frac{\dy f}{\dy y_1} dy_1 + \cdots + \frac{\dy f}{\dy y_n} dy_n = \sum_{i=1}^n\frac{\dy f}{\dy y_i}dy_i = 0,
\]
and since $X$ is smooth, the terms never all vanish at the same time. Let $V_i=\{\frac{\dy f}{\dy y_i} \neq 0\}$, and set
\[
\beta_i = \frac{(-1)^{i-1}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \in \omega_X,
\]
which is well-defined on all of $V_i\subset U_0$. We claim that the choice of $V_i$ does not matter, and indeed, assuming $i<j$,
\begin{align*}
\beta_j & = \frac{(-1)^{j-1}}{\dy f/\dy y_j} dy_1\wedge\cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{j-1+i-1}dy_i}{\dy f/\dy y_j} \wedge dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{j-1+i-1}\frac{-1}{\dy f/\dy y_i}\left(\frac{\dy f}{\dy y_1}dy_1+\cdots + \widehat{\frac{\dy f}{\dy y_i}dy_i} + \cdots + \frac{\dy f}{\dy y_n}dy_n\right)}{\dy f/\dy y_j} \wedge dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{j-1+i-1+1}\frac{1}{\dy f/\dy y_i}\cdot \frac{\dy f}{\dy y_j}dy_j}{\dy f/\dy y_j} \wedge dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{j-1+i-1+1+j-2}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{i-1}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \\
& = \beta_i.
\end{align*}
Hence $\beta_i$ is well-defined on all of $U_0$, and we call it simply $\beta$. Next we claim it is well-defined on all of $X$. Again we only check on the overlap of $U_0\cap U_n$. On the affine piece $U_n$ this becomes $g(z_1,\dots,z_n)=F(\frac{x_0}{x_n},\dots,\frac{x_{n-1}}{x_n},1)=f(\frac{z_2}{z_1},\dots,\frac{z_n}{z_1},\frac1{z_1})$ with $z_i = x_{i-1}/x_n$. We employ the chain rule $\frac{\dy f}{\dy y_i}=\frac{\dy f}{\dy z_j}\frac{\dy z_j}{\dy y_i}$ and the results above to find that
\begin{align*}
\beta & = \frac{(-1)^{i-1}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{i-1}}{\dy f/\dy z_j \cdot \dy z_j/\dy y_i} \frac{z_1dz_2-z_2dz_1}{z_1^2}\wedge \cdots \wedge \widehat{dy_i}\wedge \cdots \wedge \frac{z_1dz_n-z_ndz_1}{z_1^2}\wedge \frac{-dz_1}{z_1^2} \\
& = \frac{(-1)^{i-1}}{\dy f/\dy z_j \cdot \dy z_j/\dy y_i} \frac{(-1)^{n-1}}{z_1^n}dz_1\wedge\cdots \wedge \widehat{dz_i}\wedge \cdots \wedge dz_n \\
& = \frac{(-1)^{i+n}}{\left(\frac{1}{z_1}\right)^{d-1}\left(c+\cdots\right) z_1^n}dz_1\wedge\cdots \wedge \widehat{dz_i}\wedge \cdots \wedge dz_n \\
& = \frac{(-1)^{i+n}}{z_1^{n-d+1} \left(c+\cdots \right)}dz_1\wedge\cdots \wedge \widehat{dz_i}\wedge \cdots \wedge dz_n,
\end{align*}
for some constant $c$. This comes from expressing $f$ in terms of the $z_i$s and factoring. Since the transition function has a pole of order $n-d+1$ when $z_1 = 0$, which happens when $x_0=0$, we have that $\beta$ has a pole of order $n-d+1$ at $\infty$. Therefore $\omega_{X} \cong \mathcal O_{X}(-n+d-1)$.

References: Griffiths and Harris (Principles of Algebraic Geometry, Chapter 1.2)

The Eilenberg-Steenrod axioms

The category $\text{Top}$ of topological spaces may be generalized to the category $\text{Top}_*$ of pointed topological spaces. This in turn may be generalized to the category $\text{Top}_{rel}$ of pairs $(X,A)$, where $X\in\Obj(\text{Top})$ and $A$ is a subspace of $X$. The morphisms of $\text{Top}_{rel}$ on $(X,A)$ are the morphisms of $\text{Top}$ on $X$ paired with their restrictions to $A$. We write $(X)$ for $(X,\emptyset)$.

Definition 1: Let $X,Y\in\Obj(\text{Top}_*)$. Then $f\in\Hom_{\text{Top}_*}(X,Y)$ is an $n$-equivalence if the induced map on homotopy groups $f_*:\pi_k(X,x)\to \pi_k(Y,f(x))$ is an isomorphism for $k<n$ and an epimorphism for $k=n$. Further, $f$ is a weak equivalence if it is an $n$-equivalence for all $n\geqslant 1$. Similarly, $f\in \Hom_{\text{Top}_{rel}}((X,A),(Y,B))$ is a weak equivalence if $f\in \Hom_{\text{Top}_*}(X,Y)$ and $f|_A\in \Hom_{\text{Top}_*}(A,B)$ are weak equivalences.

Definition 2: Let $C,D$ be two categories. A functor $\mathcal F:C\to D$ is an assignment $\mathcal F(X)\in \Obj(D)$ for every $X\in \Obj(C)$, and $\mathcal F(f)\in \Hom_D(\mathcal F(X),\mathcal F(Y))$ for every $f\in\Hom_C(X,Y)$. This assignment satisfies the following relations:
          $\mathcal F(g\circ f) = \mathcal F(g)\circ \mathcal F(f)$ for every $f\in \Hom_C(X,Y)$ and $g\in \Hom_C(Y,Z)$
          $\mathcal F(\id_X) = \id_{\mathcal F(X)}$ for every $X\in\Obj(C)$

Definition 3: Let $C$ be any category and $\mathcal F:\text{Top}\to C$ a functor. Then $\mathcal F$ is homotopy invariant if $f\simeq g$ in $\text{Top}$ implies $\mathcal F(f)=\mathcal F(g)$ in $C$, where $\simeq$ is the homotopy of maps.

Definition 4: A (relative) homology theory of topological spaces is a collection of homotopy-invariant functors $H_n:\text{Top}_{rel}\to \text{Ab}$ and a collection of natural transformations $d_n:H_n(X,A) \to H_{n-1}(A)$.

The Eilenberg-Steenrod axioms are properties a relative homology theory may satisfy. The number of axioms depends on how general a view of homology theories one would like. Eilenberg and Steenrod (7), May (4), Aguilar, Gitler, and Prieto (4), Wikipedia (5), and other sources (6,8) have all different numbers of axioms. The order of the axioms below is alphabetical.

For any $(X,A)\in\Obj(\text{Top}_{rel})$ and all $n$:

Axiom 1: Additivity. If $(X,A)=\bigoplus_i(X_i,A_i)$, then $H_n(X,A) \cong \bigoplus_iH_n(X_i,A_i),$ where the isomorphism is induced by the inclusions $(X_i,A_i)\hookrightarrow (X,A)$.

Axiom 2: Exactness. There is a long exact sequence
\[ \cdots \to H_{n+1}(X,A)\tov{d_{n+1}}H_n(A)\tov{\ \ }H_n(X)\tov{\ \ }H_n(X,A)\tov{d_n}H_{n-1}(A)\tov{\ \ }\cdots \]
where $H_n(A)\to H_n(X)$ and $H_n(X)\to H_n(X,A)$ are induced by the inclusions $(A)\hookrightarrow (X)$ and $(X)\hookrightarrow (X,A)$, respectively.

Axiom 3: Excision. If there exists a subset $U$ of $X$ with $\text{cl}(U)\subset \text{int}(A)$, then there is an isomorphism $H_n(X\setminus U,A\setminus U)\cong H_n(X,A)$ induced by the inclusion $(X\setminus U,A\setminus U)\hookrightarrow (X,A)$.

Axiom 4: Dimension. $H_n(*)=0$ for all $n\neq 0$.

Axiom 5: Weak equivalence. If $f\in\Hom_{\text{Top}_{rel}}((X,A),(Y,B))$ is a weak equivalence, then the induced map on homology $f_*:H_n(X,A)\to H_n(Y,B)$ is an isomorphism.

Singular homology is a homology theory that satisfies all the axioms above. $K$-theory is a homology theory that does not satisfy the dimension axiom.

References: May (A Concise course in Algebraic Topology, Chapter 13.1), Aguilar, Gitler, and Prieto (Algebraic Topology from a Homotopical Viewpoint, Chapter 5.3)