Topologists had been studying homotopy theory long before we conceived of homotopical algebra; the definition of model category is abstracting how they did so.

The modern philosophy is that homotopical algebra is $\infty$-category theory (which is short for $(\infty, 1)$-category). In practice, this is often done by reducing questions of $\infty$-category theory to questions of 1-category theory (i.e. the usual category theory).

A particularly common form of doing this is to sandwich an $\infty$ category $\mathcal{C}$ between two 1-categories $C$ and $h\mathcal{C}$ with functors

$$ C \xrightarrow{L} \mathcal{C} \xrightarrow{\pi} h\mathcal{C}$$

such that this arrangement has the following properties. Let $W \subseteq C$ be the subcategory of arrows $w$ for which $\pi L(w)$ is an isomorphism.

• If $w$ is an arrow of $C$, then $w \in W$ if and only if $L(w)$ is an equivalence.
• If $D$ is a 1-category and $F : \mathcal{C} \to D$ is a functor, then there is a unique¹ functor $G : h\mathcal{C} \to D$ such that $F \simeq G \pi$.
• If $\mathcal{D}$ is an $\infty$-category and $F : C \to \mathcal{D}$ is a functor such $F(w)$ is an equivalence for every $w \in W$, then there is a unique¹ functor $G : \mathcal{C} \to \mathcal{D}$ such that $F = G L$.

_{1: up to unique¹ equivalence}

In short, $\pi : \mathcal{C} \to h\mathcal{C}$ is universal among arrows from $\mathcal{C}$ to 1-categories, and $L : C \to \mathcal{C}$ expresses $\mathcal{C}$ identifies $\mathcal{C}$ as the localization of $C$ obtained by turning the arrows of $W$ into equivalences.

The pair $(C, W)$ is called a saturated relative category, or a saturated homotopical category. The category $h\mathcal{C}$ is called the homotopy category of $\mathcal{C}$.

So, the functor $C \to h \mathcal{C}$ gives an avatar in 1-category theory for the $\infty$-category $\mathcal{C}$. Furthermore:

• We can identify which such functors are avatars of $\infty$-categories: they're the ones for which $h\mathcal{C}$ is the localization of $C$ by making everything in $W$ an isomorphism
• Every $\infty$ category can be expressed in this manner

and so this gives a way to work with $\infty$-categories in general, entirely in the settings of 1-category theory. But we're greedy: we don't want to just have avatars of the categories, we also want to reduce everything we can to 1-category theory.

Model categories are extra structure that lets us do a lot of that. The fibrations and cofibrations satisfying the weak factorization axioms give us algebraic tools for reducing questions about $\mathcal{C}$ to the corresponding question about $C$.

For example, if you have a diagram $F : J \to C$ where every vertex is fibrant and every arrow is a fibration, then $L(\lim F) \simeq \lim(LF)$: that is, you can compute the limit in $\mathcal{C}$ simply by computing the limit in $C$. By "fibrant replacement", you can reduce any diagram $J \to \mathcal{C}$ to one of the above form, and so this gives us a way to compute limits in $\mathcal{C}$ by computing limits in $C$. (at least, when the index category is a $1$-category)

Topologists had been studying homotopy theory long before we conceived of homotopical algebra; the definition of model category is abstracting how they did so.

The modern philosophy is that homotopical algebra is $\infty$-category theory (which is short for $(\infty, 1)$-category). In practice, this is often done by reducing questions of $\infty$-category theory to questions of 1-category theory (i.e. the usual category theory).

A particularly common form of doing this is to sandwich an $\infty$ category $\mathcal{C}$ between two 1-categories $C$ and $h\mathcal{C}$ with functors

$$ C \xrightarrow{L} \mathcal{C} \xrightarrow{\pi} h\mathcal{C}$$

such that this arrangement has the following properties. Let $W \subseteq C$ be the subcategory of arrows $w$ for which $\pi L(w)$ is an isomorphism.

• If $w$ is an arrow of $C$, then $w \in W$ if and only if $L(w)$ is an equivalence.
• If $D$ is a 1-category and $F : \mathcal{C} \to D$ is a functor, then there is a unique¹ functor $G : h\mathcal{C} \to D$ such that $F \simeq G \pi$.
• If $\mathcal{D}$ is an $\infty$-category and $F : C \to \mathcal{D}$ is a functor such $F(w)$ is an equivalence for every $w \in W$, then there is a unique¹ functor $G : \mathcal{C} \to \mathcal{D}$ such that $F = G L$.

_{1: up to unique¹ equivalence}

In short, $\pi : \mathcal{C} \to h\mathcal{C}$ is universal among arrows from $\mathcal{C}$ to 1-categories, and $L : C \to \mathcal{C}$ expresses $\mathcal{C}$ identifies $\mathcal{C}$ as the localization of $C$ obtained by turning the arrows of $W$ into equivalences.

The pair $(C, W)$ is called a saturated relative category, or a saturated homotopical category. The category $h\mathcal{C}$ is called the homotopy category of $\mathcal{C}$.

So, the functor $C \to h \mathcal{C}$ gives an avatar in 1-category theory for the $\infty$-category $\mathcal{C}$. Furthermore:

• We can identify which such functors are avatars of $\infty$-categories: they're the ones for which $h\mathcal{C}$ is the localization of $C$ by making everything in $W$ an isomorphism
• Every $\infty$ category can be expressed in this manner

and so this gives a way to work with $\infty$-categories in general, entirely in the settings of 1-category theory. Note that one usually specifies the pair $(C, W)$ rather than the functor $C \to h \mathcal{C}$. (and often weakens the condition on $W$)

However, we're greedy: we don't want to just have avatars of the categories, we also want to reduce everything we can to 1-category theory.

Model categories are extra structure that lets us do a lot of that. The fibrations and cofibrations satisfying the weak factorization axioms give us algebraic tools for reducing questions about $\mathcal{C}$ to the corresponding question about $C$.

For example, if you have a diagram $F : J \to C$ where every vertex is fibrant and every arrow is a fibration, then $L(\lim F) \simeq \lim(LF)$: that is, you can compute the limit in $\mathcal{C}$ simply by computing the limit in $C$. By "fibrant replacement", you can reduce any diagram $J \to \mathcal{C}$ to one of the above form, and so this gives us a way to compute limits in $\mathcal{C}$ by computing limits in $C$. (at least, when the index category is a $1$-category)

Topologists had been studying homotopy theory long before we conceived of homotopical algebra; the definition of model category is abstracting how they did so.

The modern philosophy is that homotopical algebra is $\infty$-category theory (which is short for $(\infty, 1)$-category). In practice, this is often done by reducing questions of $\infty$-category theory to questions of 1-category theory (i.e. the usual category theory).

A particularly common form of doing this is to sandwich an $\infty$ category $\mathcal{C}$ between two 1-categories $C$ and $h\mathcal{C}$ with functors

$$ C \xrightarrow{L} \mathcal{C} \xrightarrow{\pi} h\mathcal{C}$$

such that this arrangement has the following properties. Let $W \subseteq C$ be the subcategory of arrows $w$ for which $\pi L(w)$ is an isomorphism.

• If $w$ is an arrow of $C$, then $w \in W$ if and only if $L(w)$ is an equivalence.
• If $D$ is a 1-category and $F : \mathcal{C} \to D$ is a functor, then there is a unique¹ functor $G : h\mathcal{C} \to D$ such that $F \simeq G \pi$.
• If $\mathcal{D}$ is an $\infty$-category and $F : C \to \mathcal{D}$ is a functor such $F(w)$ is an equivalence for every $w \in W$, then there is a unique¹ functor $G : \mathcal{C} \to \mathcal{D}$ such that $F = G L$.

_{1: up to unique¹ equivalence}

In short, $\pi : \mathcal{C} \to h\mathcal{C}$ is universal among arrows from $\mathcal{C}$ to 1-categories, and $L : C \to \mathcal{C}$ expresses $\mathcal{C}$ identifies $\mathcal{C}$ as the localization of $C$ obtained by turning the arrows of $W$ into equivalences.

The pair $(C, W)$ is called a saturated relative category, or a saturated homotopical category. The category $h\mathcal{C}$ is called the homotopy category of $\mathcal{C}$.

It turns out that you can do this for every $\infty$-category, so this at least gives us avatars of $\infty$-categories in the setting of 1-category theory. But we're greedy: we want to reduce everything we can to 1-category theory.

Model categories are extra structure that lets us do a lot of that. The fibrations and cofibrations satisfying the weak factorization axioms give us algebraic tools for reducing questions about $\mathcal{C}$ to the corresponding question about $C$.

For example, if you have a diagram $F : J \to C$ where every vertex is fibrant and every arrow is a fibration, then $L(\lim F) \simeq \lim(LF)$: that is, you can compute the limit in $\mathcal{C}$ simply by computing the limit in $C$. By "fibrant replacement", you can reduce any diagram $J \to \mathcal{C}$ to one of the above form, and so this gives us a way to compute limits in $\mathcal{C}$ by computing limits in $C$. (at least, when the index category is a $1$-category)

Topologists had been studying homotopy theory long before we conceived of homotopical algebra; the definition of model category is abstracting how they did so.

The modern philosophy is that homotopical algebra is $\infty$-category theory (which is short for $(\infty, 1)$-category). In practice, this is often done by reducing questions of $\infty$-category theory to questions of 1-category theory (i.e. the usual category theory).

A particularly common form of doing this is to sandwich an $\infty$ category $\mathcal{C}$ between two 1-categories $C$ and $h\mathcal{C}$ with functors

$$ C \xrightarrow{L} \mathcal{C} \xrightarrow{\pi} h\mathcal{C}$$

such that this arrangement has the following properties. Let $W \subseteq C$ be the subcategory of arrows $w$ for which $\pi L(w)$ is an isomorphism.

• If $w$ is an arrow of $C$, then $w \in W$ if and only if $L(w)$ is an equivalence.
• If $D$ is a 1-category and $F : \mathcal{C} \to D$ is a functor, then there is a unique¹ functor $G : h\mathcal{C} \to D$ such that $F \simeq G \pi$.
• If $\mathcal{D}$ is an $\infty$-category and $F : C \to \mathcal{D}$ is a functor such $F(w)$ is an equivalence for every $w \in W$, then there is a unique¹ functor $G : \mathcal{C} \to \mathcal{D}$ such that $F = G L$.

_{1: up to unique¹ equivalence}

In short, $\pi : \mathcal{C} \to h\mathcal{C}$ is universal among arrows from $\mathcal{C}$ to 1-categories, and $L : C \to \mathcal{C}$ expresses $\mathcal{C}$ identifies $\mathcal{C}$ as the localization of $C$ obtained by turning the arrows of $W$ into equivalences.

The pair $(C, W)$ is called a saturated relative category, or a saturated homotopical category. The category $h\mathcal{C}$ is called the homotopy category of $\mathcal{C}$.

So, the functor $C \to h \mathcal{C}$ gives an avatar in 1-category theory for the $\infty$-category $\mathcal{C}$. Furthermore:

• We can identify which such functors are avatars of $\infty$-categories: they're the ones for which $h\mathcal{C}$ is the localization of $C$ by making everything in $W$ an isomorphism
• Every $\infty$ category can be expressed in this manner

and so this gives a way to work with $\infty$-categories in general, entirely in the settings of 1-category theory. But we're greedy: we don't want to just have avatars of the categories, we also want to reduce everything we can to 1-category theory.

Model categories are extra structure that lets us do a lot of that. The fibrations and cofibrations satisfying the weak factorization axioms give us algebraic tools for reducing questions about $\mathcal{C}$ to the corresponding question about $C$.

For example, if you have a diagram $F : J \to C$ where every vertex is fibrant and every arrow is a fibration, then $L(\lim F) \simeq \lim(LF)$: that is, you can compute the limit in $\mathcal{C}$ simply by computing the limit in $C$. By "fibrant replacement", you can reduce any diagram $J \to \mathcal{C}$ to one of the above form, and so this gives us a way to compute limits in $\mathcal{C}$ by computing limits in $C$. (at least, when the index category is a $1$-category)