Lecture 1

Definition 1.1.1. Given a commutative square of the following form:

a lift in the diagram is a morphism $h \colon B \to X$ such that $hi=f$ and $ph=g$. A morphism $i \colon A \to B$ is said to have the left lifting property (LLP) with respect to another morphism $p \colon X \to Y$ and $p$ is said to have the right lifting property (RLP) with respect to $i$ if a lifting exists for any choice of $f$ and $g$ making the above diagram commute.

Definition 1.1.2. Let $f \colon X \to Y$ be a morphism in a category $\mathcal{C}$, then a retraction of $f$ is left-inverse. That is, there exists a morphism $r$ such that $r \circ f = \mathrm{id}_ X$.

Definition 1.1.3. A model category is a category $\mathcal{C}$ with three distinguished classes of morphisms:

• Weak equivalences - $\mathrm{W}_\mathcal {C}$;
• Fibrations - $\mathrm{Fib}_\mathcal {C}$;
• Cofibrations - $\mathrm{Cof}_\mathcal {C}$;

• (MC1) $\mathcal{C}$ has all small limits and colimits. In particular, there is an initial object $\emptyset $ and a terminal object $\ast $.
• (MC2) If $f$ and $g$ are morphism such that $gf$ is defined and if two of $f$, $g$ and $gf$ are weak equivalences, then so is the third. That is, the weak equivalences satisfy the 2-out-of-3 property.
• (MC3) The three distinguished classes of morphisms are closed under retracts.
• (MC4) Given a commutative diagram of the form above, a lift exists when either $i$ is a cofibration and $p$ is an acyclic fibration or when $i$ is an acyclic cofibration and $p$ is a fibration.
• (MC5) Each morphism $f$ in $\mathcal{C}$ can be factored in two ways:
  1. $f = pi$, where $i$ is a cofibration and $p$ is an acyclic fibration.
  2. $f = pi$, where $p$ is a fibration and $i$ is an acyclic cofibration.

Definition 1.1.5. Let $\mathcal{C}$ be a model category. An object $X \in \mathcal{C}$ is said to be:

• fibrant if the unique morphism $X \to \ast $ is a fibration;
• cofibrant is the unique morphism $\emptyset \to X$ is a cofibration;
• bifibrant if it is both fibrant and cofibrant.

Definition 1.1.6. Let $\mathcal{C}$ be a model category and $X \in \mathcal{C}$.

• A fibrant replacement of $X$ is an object $RX$ equipped with an acyclic cofibration $X \to RX$ such that $RX$ is fibrant.
• A cofibrant replacement of $X$ is an object $QX$ along with an acyclic fibration $QX \to X$ such that $QX$ is cofibrant.

Definition 1.2.1. Let $\mathcal{C}$ be a model category and $X \in \mathcal{C}$. A cylinder object $\operatorname {Cyl}(X)$ for $X$ is a factorization of the codiagonal $\nabla _ X \colon X \sqcup X \to X$ as

Note that such a factorization exists by MC5.

Definition 1.2.2. Let $f,g \colon X \to Y$ be a pair of morphisms in a model category. Then a left homotopy $\eta \colon f \sim _ L g$ is a morphism $\eta \colon \operatorname {Cyl}(X) \to Y$ which makes the following diagram commute:

Definition 1.2.3. Let $\mathcal{C}$ be a model category and $X \in \mathcal{C}$. A path object $\operatorname {Path}(X)$ for $X$ is a factorization of the diagonal $\Delta _ X \colon X\to X \times X$ as

Note that such a factorization exists by MC5

Definition 1.2.4. Let $f,g \colon X \to Y$ be a pair of morphisms in a model category. Then a right homotopy $\eta \colon f \sim _ R g$ is a morphism $\eta \colon X \to \operatorname {Path}(Y)$ which makes the following diagram commute:

Definition 1.2.5.

1. A pair of morphisms $f,g \colon X \to Y$ in a model category are homotopic, written $f \sim g$ if they are both left and right homotopic.
2. A morphism $f \colon X \to Y$ in a model category is a homotopy equivalence is there is a morphism $h \colon Y \to X$ such that $hf \sim \mathrm{id}_ X$ and $fh \sim \mathrm{id}_ Y$.

Definition 1.2.8. Let $\mathcal{C}$ be a model category. Then its homotopy category $\operatorname {Ho}(\mathcal{C})$, up to equivalence of categories, is the category whose

• objects are the objects of $\mathcal{C}_{cf}$;
• morphisms are the homotopy classes of morphisms in $\mathcal{C}$. That is, the equivalence class of morphisms under homotopy.

Definition 1.2.9. Let $\mathcal{C}$ be a category equipped with a class of weak equivalences. The object $\mathcal{C}[W^{-1}]$ is constructed in the following fashion. Begin by forming the free category $F(\mathcal{C},W^{-1})$ on the morphisms of $\mathcal{C}$ and the reversals of the morphisms of $W$. In particular an object of $F(\mathcal{C},W^{-1})$ is an object of $\mathcal{C}$ and a morphism is a finite string of composable morphisms $(f_1, f_2, \dots , f_ n)$ where $f_ i$ is either a morphism of $\mathcal{C}$ or the reversal $w_ i^{-1}$ of a morphism $w_ i \in W$. The empty string at a particular object plays the role of the identity at that object, and the composition is given by concatenation of strings.

One then defines $\mathcal{C}[W^{-1}]$ to be the quotient category of $F(\mathcal{C},W^{-1})$ by the following relations:

• $\operatorname {id}_ A = (\operatorname {id}_ A)$ for all objects $A$;
• $(f,g) = (g \circ f)$ for all composable morphisms $f,g$ of $\mathcal{C}$;
• $\operatorname {id}_{\operatorname {dom}w} = (w,w^{-1})$ for all $w \in W$;
• $\operatorname {id}_{\operatorname {codom}w} = (w^{-1},w)$ for all $w \in W$.

Definition 1.3.2. A map $f \colon X \to Y$ of topological spaces is said to be a weak homotopy equivalence if for each point $x \in X$ the map $f_\ast \colon \pi _ n (X,x) \to \pi _ n(Y,f(x))$ is a bijection of pointed sets for $n=0$, and an isomorphism of groups for $n \geq 1$.

Definition 1.3.3. A map $f \colon X \to Y$ of topological spaces is a Serre fibration if for each CW-complex $A$, the map $f$ has the RLP with respect to the inclusion $A \times 0 \to A \times [0,1]$.

Definition 1.3.8.

• A functor $F \colon \mathcal{C} \to \mathcal{D}$ is an equivalence if it is fully faithful and essentially surjective.
• A functor $F \colon \mathcal{C} \to \mathcal{D}$ is an isofibration for all $c \in \mathcal{C}$ and any isomorphism $\phi \colon F(c) \cong d$, there exists an isomorphism $\psi \colon c \cong c'$ such that $F(\psi ) = \phi $.