Model Categories through Lifting I: From Binary Relations to Model Categories

Model Categories through Lifting I: From Binary Relations to Model CategoriesThe guiding problemBinary relations and Galois connectionsAntitonicityClosure operatorsClosed classesThe lifting relationRLP and LLPConsequences for RLP and LLPClosed lifting pairsWeak factorization systemsFunctorial factorizationFormal consequences of a weak factorization systemModel categoriesCofibrant and fibrant objectsCofibrant replacementFibrant replacementBifibrant replacementWhy replacements matterConceptual summary

Model Categories through Lifting I: From Binary Relations to Model Categories

The guiding problem

The basic problem behind model categories is the following.

We often have a category $\mathcal{C}$$\mathcal C$ together with a class of morphisms $W$$W$ which we want to regard as equivalences. These morphisms may not be isomorphisms inside $\mathcal{C}$$\mathcal C$, but we want them to become isomorphisms after passing to a new category.

Formally, one can try to construct a localization

$$\mathcal{C}[W^{-1}].$$

This means that every morphism in $W$$W$ is formally inverted.

However, this localization is often hard to compute directly. Morphisms in $\mathcal{C}[W^{-1}]$$\mathcal C[W^{-1}]$ may be represented by zigzags, and it is usually not obvious how to work with them concretely.

A model category structure is extra structure on $\mathcal{C}$$\mathcal C$ which makes this localization computable.

It does this by adding two further classes of morphisms:

$$\mathrm{Cof}$$

and

$$\mathrm{Fib}.$$

These are called cofibrations and fibrations.

The point is not merely that we have three classes of morphisms. The point is that these classes are organized by two weak factorization systems:

$$(\mathrm{Cof}\cap W,\mathrm{Fib})$$

and

$$(\mathrm{Cof},\mathrm{Fib}\cap W).$$

Thus a model category should be understood as:

$$\text{weak equivalences}+\text{two compatible weak factorization systems}.$$

The class $W$$W$ tells us what should become invertible.

The two weak factorization systems tell us how to replace objects by better ones before computing.

This article explains the formal structure behind this idea.

Binary relations and Galois connections

The formal mechanism behind RLP and LLP is not specific to model categories. It already exists for any binary relation.

Let $P$$P$ and $Q$$Q$ be sets, and let

$$\mathcal{R}\subseteq P\times Q$$

be a binary relation.

For a subset $A\subseteq P$$A\subseteq P$, define

$$\mathrm{\Phi }(A)=\{q\in Q\mid \text{for every }p\in A,(p,q)\in \mathcal{R}\}.$$

For a subset $B\subseteq Q$$B\subseteq Q$, define

$$\mathrm{\Psi }(B)=\{p\in P\mid \text{for every }q\in B,(p,q)\in \mathcal{R}\}.$$

Thus $\mathrm{\Phi }(A)$$\Phi(A)$ consists of all elements of $Q$$Q$ related to every element of $A$$A$, while $\mathrm{\Psi }(B)$$\Psi(B)$ consists of all elements of $P$$P$ related to every element of $B$$B$.

The key observation is that $\mathrm{\Phi }$$\Phi$ and $\mathrm{\Psi }$$\Psi$ form a contravariant Galois connection:

$$A\subseteq \mathrm{\Psi }(B)⟺B\subseteq \mathrm{\Phi }(A).$$

Indeed, the left-hand side says

$$\mathrm{\forall }p\in A,p\in \mathrm{\Psi }(B).$$

By definition of $\mathrm{\Psi }(B)$$\Psi(B)$, this means

$$\mathrm{\forall }p\in A,\mathrm{\forall }q\in B,(p,q)\in \mathcal{R}.$$

The right-hand side says

$$\mathrm{\forall }q\in B,q\in \mathrm{\Phi }(A).$$

By definition of $\mathrm{\Phi }(A)$$\Phi(A)$, this means

$$\mathrm{\forall }q\in B,\mathrm{\forall }p\in A,(p,q)\in \mathcal{R}.$$

These two statements are equivalent, since they differ only by the order of two universal quantifiers.

Therefore

$$A\subseteq \mathrm{\Psi }(B)⟺B\subseteq \mathrm{\Phi }(A).$$

This is the formal pattern behind RLP and LLP.

Antitonicity

Both operations are order-reversing.

If

$$A\subseteq A^{\prime },$$

then being related to every element of $A^{\prime }$$A'$ is a stronger condition than being related to every element of $A$$A$. Hence

$$\mathrm{\Phi }(A^{\prime })\subseteq \mathrm{\Phi }(A).$$

Similarly, if

$$B\subseteq B^{\prime },$$

then

$$\mathrm{\Psi }(B^{\prime })\subseteq \mathrm{\Psi }(B).$$

Thus enlarging the test class makes the class of objects satisfying all tests smaller.

Closure operators

Although $\mathrm{\Phi }$$\Phi$ and $\mathrm{\Psi }$$\Psi$ are order-reversing, their composites

$$\mathrm{\Psi }\mathrm{\Phi }:\mathcal{P}(P)\to \mathcal{P}(P)$$

and

$$\mathrm{\Phi }\mathrm{\Psi }:\mathcal{P}(Q)\to \mathcal{P}(Q)$$

are order-preserving.

Moreover, they are closure operators.

First, for every $A\subseteq P$$A\subseteq P$, we have

$$A\subseteq \mathrm{\Psi }\mathrm{\Phi }(A).$$

Indeed, by the Galois connection, this is equivalent to

$$\mathrm{\Phi }(A)\subseteq \mathrm{\Phi }(A),$$

which is tautological.

Similarly, for every $B\subseteq Q$$B\subseteq Q$,

$$B\subseteq \mathrm{\Phi }\mathrm{\Psi }(B).$$

Second, these closure operators are idempotent. Equivalently, we have the triple identities

$$\mathrm{\Phi }\mathrm{\Psi }\mathrm{\Phi }=\mathrm{\Phi }$$

and

$$\mathrm{\Psi }\mathrm{\Phi }\mathrm{\Psi }=\mathrm{\Psi }.$$

For example, since

$$A\subseteq \mathrm{\Psi }\mathrm{\Phi }(A),$$

and $\mathrm{\Phi }$$\Phi$ is order-reversing, we get

$$\mathrm{\Phi }\mathrm{\Psi }\mathrm{\Phi }(A)\subseteq \mathrm{\Phi }(A).$$

On the other hand, applying the extensivity statement to $B=\mathrm{\Phi }(A)$$B=\Phi(A)$ gives

$$\mathrm{\Phi }(A)\subseteq \mathrm{\Phi }\mathrm{\Psi }\mathrm{\Phi }(A).$$

Therefore

$$\mathrm{\Phi }\mathrm{\Psi }\mathrm{\Phi }(A)=\mathrm{\Phi }(A).$$

The proof of

$$\mathrm{\Psi }\mathrm{\Phi }\mathrm{\Psi }(B)=\mathrm{\Psi }(B)$$

is dual.

Closed classes

A subset $A\subseteq P$$A\subseteq P$ is called closed if

$$A=\mathrm{\Psi }\mathrm{\Phi }(A).$$

A subset $B\subseteq Q$$B\subseteq Q$ is called closed if

$$B=\mathrm{\Phi }\mathrm{\Psi }(B).$$

The closed subsets of $P$$P$ are precisely the subsets in the image of $\mathrm{\Psi }$$\Psi$.

Indeed, every subset of the form $\mathrm{\Psi }(B)$$\Psi(B)$ is closed, because

$$\mathrm{\Psi }\mathrm{\Phi }\mathrm{\Psi }(B)=\mathrm{\Psi }(B).$$

Conversely, if $A$$A$ is closed, then

$$A=\mathrm{\Psi }\mathrm{\Phi }(A),$$

so $A$$A$ lies in the image of $\mathrm{\Psi }$$\Psi$.

Similarly, the closed subsets of $Q$$Q$ are precisely the subsets in the image of $\mathrm{\Phi }$$\Phi$.

Therefore $\mathrm{\Phi }$$\Phi$ and $\mathrm{\Psi }$$\Psi$ restrict to mutually inverse order-reversing bijections between their images:

$$\mathrm{\Phi }:\mathrm{Im}(\mathrm{\Psi })\to \mathrm{Im}(\mathrm{\Phi }),$$

and

$$\mathrm{\Psi }:\mathrm{Im}(\mathrm{\Phi })\to \mathrm{Im}(\mathrm{\Psi }).$$

This is the general Galois-theoretic mechanism behind closed lifting classes.

The lifting relation

Now let $\mathcal{C}$$\mathcal C$ be a category.

We specialize the above construction to the case

$$P=Q=\mathrm{Mor}(\mathcal{C}).$$

The binary relation is the lifting relation.

Let

$$i:A\to B$$

and

$$p:X\to Y$$

be morphisms in $\mathcal{C}$$\mathcal C$.

We say that $p$$p$ has the right lifting property with respect to $i$$i$ if every commutative square

$$\begin{array}{ccc}A& \stackrel{a}{\to }& X\\ {\downarrow }^i& & {\downarrow }^p\\ B& \stackrel{b}{\to }& Y\end{array}$$

admits a diagonal filler

$$h:B\to X$$

such that

$$hi=a$$

and

$$ph=b.$$

Equivalently, every lifting problem of this form has at least one solution.

Dually, we say that $i$$i$ has the left lifting property with respect to $p$$p$ if $p$$p$ has the right lifting property with respect to $i$$i$.

This is an existence condition. It says that compatible boundary data can be extended across the square. It does not require uniqueness of the lift.

RLP and LLP

For a class of morphisms

$$J\subseteq \mathrm{Mor}(\mathcal{C}),$$

define

$$\mathrm{RLP}(J)=\{p\in \mathrm{Mor}(\mathcal{C})\mid p\text{ has the right lifting property with respect to every }j\in J\}.$$

Similarly,

$$\mathrm{LLP}(J)=\{i\in \mathrm{Mor}(\mathcal{C})\mid i\text{ has the left lifting property with respect to every }j\in J\}.$$

These are exactly the two operations arising from the lifting relation.

For two classes of morphisms

$$S,T\subseteq \mathrm{Mor}(\mathcal{C}),$$

we have

$$S\subseteq \mathrm{LLP}(T)⟺T\subseteq \mathrm{RLP}(S).$$

Indeed, the left-hand side says:

for every $s\in S$$s\in S$ and every $t\in T$$t\in T$, the morphism $s$$s$ has the left lifting property with respect to $t$$t$.

The right-hand side says:

for every $t\in T$$t\in T$ and every $s\in S$$s\in S$, the morphism $t$$t$ has the right lifting property with respect to $s$$s$.

These are the same statement.

Thus RLP and LLP form a Galois connection.

Consequences for RLP and LLP

Since RLP and LLP arise from a Galois connection, both operations are order-reversing.

If

$$S\subseteq S^{\prime },$$

then

$$\mathrm{RLP}(S^{\prime })\subseteq \mathrm{RLP}(S).$$

If

$$T\subseteq T^{\prime },$$

then

$$\mathrm{LLP}(T^{\prime })\subseteq \mathrm{LLP}(T).$$

We also obtain the triple identities

$$\mathrm{RLP}(\mathrm{LLP}(\mathrm{RLP}(J)))=\mathrm{RLP}(J)$$

and

$$\mathrm{LLP}(\mathrm{RLP}(\mathrm{LLP}(J)))=\mathrm{LLP}(J).$$

For example, set

$$R=\mathrm{RLP}(J).$$

Then

$$\mathrm{LLP}(R)=\mathrm{LLP}(\mathrm{RLP}(J)).$$

Applying $\mathrm{RLP}$$\operatorname{RLP}$ again gives

$$\mathrm{RLP}(\mathrm{LLP}(R))=R.$$

Hence

$$\mathrm{RLP}(\mathrm{LLP}(\mathrm{RLP}(J)))=\mathrm{RLP}(J).$$

This identity is not a special theorem about model categories. It is a formal consequence of the Galois connection associated to the lifting relation.

Closed lifting pairs

A class of morphisms $R$$R$ is right-closed if

$$R=\mathrm{RLP}(\mathrm{LLP}(R)).$$

Equivalently, $R$$R$ lies in the image of $\mathrm{RLP}$$\operatorname{RLP}$.

A class of morphisms $L$$L$ is left-closed if

$$L=\mathrm{LLP}(\mathrm{RLP}(L)).$$

Equivalently, $L$$L$ lies in the image of $\mathrm{LLP}$$\operatorname{LLP}$.

The Galois connection gives an order-reversing bijection between left-closed and right-closed classes:

$$L⟼\mathrm{RLP}(L),$$

and

$$R⟼\mathrm{LLP}(R).$$

Thus a pair of classes

$$L,R\subseteq \mathrm{Mor}(\mathcal{C})$$

satisfying

$$L=\mathrm{LLP}(R)$$

and

$$R=\mathrm{RLP}(L)$$

is a closed pair for the lifting Galois connection.

This is the formal core of weak factorization systems.

Weak factorization systems

A weak factorization system is more than a closed lifting pair.

Let $\mathcal{C}$$\mathcal C$ be a category, and let $L$$L$ and $R$$R$ be two classes of morphisms in $\mathcal{C}$$\mathcal C$.

A weak factorization system consists of $L$$L$ and $R$$R$ such that

$$L=\mathrm{LLP}(R),$$

$$R=\mathrm{RLP}(L),$$

and every morphism

$$f:A\to B$$

admits a functorial factorization

$$A\stackrel{l_f}{\to }E(f)\stackrel{r_f}{\to }B$$

with

$$l_f\in L,$$

$$r_f\in R,$$

and

$$r_f{l}_f=f.$$

The first two conditions say that $L$$L$ and $R$$R$ determine each other by lifting properties.

The third condition says that every morphism can actually be decomposed using these two classes.

Thus:

$$\text{weak factorization system}=\text{closed lifting pair}+\text{factorization of every morphism}.$$

The closed lifting pair comes from the Galois connection.

The factorization condition is additional.

Functorial factorization

The word "functorial" means that the factorization is not chosen independently for each morphism.

Consider two morphisms

$$f:A\to B$$

and

$$g:A^{\prime }\to B^{\prime }.$$

A morphism from $f$$f$ to $g$$g$ in the arrow category is a commutative square

$$\begin{array}{ccc}A& \stackrel{u}{\to }& A^{\prime }\\ {\downarrow }^f& & {\downarrow }^g\\ B& \stackrel{v}{\to }& B^{\prime }\end{array}$$

meaning that

$$gu=vf.$$

Suppose $f$$f$ and $g$$g$ are factored as

$$A\stackrel{l_f}{\to }E(f)\stackrel{r_f}{\to }B$$

and

$$A^{\prime }\stackrel{l_g}{\to }E(g)\stackrel{r_g}{\to }{B}^{\prime }.$$

Functoriality requires that the commutative square from $f$$f$ to $g$$g$ induce a morphism

$$E(f)\to E(g)$$

such that the diagram

$$\begin{array}{ccccc}A& \stackrel{l_f}{\to }& E(f)& \stackrel{r_f}{\to }& B\\ {\downarrow }^u& & \downarrow & & {\downarrow }^v\\ A^{\prime }& \stackrel{l_g}{\to }& E(g)& \stackrel{r_g}{\to }& B^{\prime }\end{array}$$

commutes.

Equivalently, the induced morphism $E(f)\to E(g)$$E(f)\to E(g)$ satisfies

$$E(f)\to E(g)\text{ composed with }{l}_f=l_{g}u,$$

and

$$r_g\text{ composed with }E(f)\to E(g)=v{r}_f.$$

It must also respect identity squares and composition of squares.

So a functorial factorization is a coherent global choice of factorizations.

This is important because replacement procedures then become functors. For example, in a model category, cofibrant replacement is obtained by factoring

$$\varnothing \to X.$$

If the factorization is functorial, then a morphism

$$X\to Y$$

automatically induces a morphism

$$QX\to QY.$$

Thus $Q$$Q$ becomes a functor.

The same applies to fibrant replacement.

Formal consequences of a weak factorization system

The lifting description immediately gives useful closure properties.

In a weak factorization system $(L,R)$$(L,R)$:

• every isomorphism lies in both $L$$L$ and $R$$R$;

• both $L$$L$ and $R$$R$ are closed under composition;

• $L$$L$ is closed under pushouts;

• $R$$R$ is closed under pullbacks;

• both $L$$L$ and $R$$R$ are closed under retracts.

These are not extra axioms. They follow from the lifting definitions and the retract argument.

For example, $L$$L$ is closed under pushouts because a pushout of a map with a left lifting property still has the same left lifting property. Dually, $R$$R$ is closed under pullbacks.

The closure under composition is especially useful when proving that iterated replacements remain cofibrant or fibrant.

Model categories

We now add weak equivalences.

Let $\mathcal{C}$$\mathcal C$ be a category with all small limits and colimits.

A model category structure on $\mathcal{C}$$\mathcal C$ consists of three distinguished classes of morphisms

$$W,\mathrm{Cof},\mathrm{Fib}\subseteq \mathrm{Mor}(\mathcal{C}),$$

called weak equivalences, cofibrations, and fibrations, such that:

1. $W$$W$ contains all isomorphisms, is closed under retracts, and satisfies the two-out-of-three property;

2. the pair

$$(\mathrm{Cof}\cap W,\mathrm{Fib})$$

is a weak factorization system;

3. the pair

$$(\mathrm{Cof},\mathrm{Fib}\cap W)$$

is a weak factorization system.

Here

$$\mathrm{Cof}\cap W$$

is the class of trivial cofibrations, and

$$\mathrm{Fib}\cap W$$

is the class of trivial fibrations.

The word "trivial" means "also a weak equivalence." It does not mean "zero" or "unimportant."

The two weak factorization systems say that every morphism can be factored in two different ways.

Using

$$(\mathrm{Cof}\cap W,\mathrm{Fib}),$$

every morphism factors as

$$\text{trivial cofibration}\text{followed by}\text{fibration}.$$

Using

$$(\mathrm{Cof},\mathrm{Fib}\cap W),$$

every morphism factors as

$$\text{cofibration}\text{followed by}\text{trivial fibration}.$$

The lifting identities are:

$$\mathrm{Cof}\cap W=\mathrm{LLP}(\mathrm{Fib}),$$

$$\mathrm{Fib}=\mathrm{RLP}(\mathrm{Cof}\cap W),$$

and

$$\mathrm{Cof}=\mathrm{LLP}(\mathrm{Fib}\cap W),$$

$$\mathrm{Fib}\cap W=\mathrm{RLP}(\mathrm{Cof}).$$

Thus a model category is not just a category with three unrelated classes of morphisms. It is a weak equivalence category equipped with two compatible weak factorization systems.

The slogan is:

$$\text{weak equivalences tell us what to invert;}$$

$$\text{the two weak factorization systems tell us how to compute after inverting them.}$$

Cofibrant and fibrant objects

Assume $\mathcal{C}$$\mathcal C$ has an initial object $\varnothing$$\varnothing$ and a terminal object $\ast$$*$.

An object $X$$X$ is called cofibrant if the canonical morphism

$$\varnothing \to X$$

is a cofibration.

An object $X$$X$ is called fibrant if the canonical morphism

$$X\to \ast$$

is a fibration.

Thus cofibrancy is measured by the map out of the initial object, while fibrancy is measured by the map into the terminal object.

These notions depend on the chosen model structure.

The same underlying object may be cofibrant in one model structure and not cofibrant in another.

Cofibrant replacement

Let $X$$X$ be an object of a model category.

Consider the canonical morphism

$$\varnothing \to X.$$

Using the weak factorization system

$$(\mathrm{Cof},\mathrm{Fib}\cap W),$$

we factor it as

$$\varnothing ⟶QX⟶X,$$

where

$$\varnothing \to QX$$

is a cofibration, and

$$QX\to X$$

is a trivial fibration.

Since

$$\varnothing \to QX$$

is a cofibration, $QX$$QX$ is cofibrant.

Since

$$QX\to X$$

is a weak equivalence, $QX$$QX$ is weakly equivalent to $X$$X$.

Thus

$$QX\to X$$

is called a cofibrant replacement of $X$$X$.

The diagram is:

$$\varnothing \stackrel{\mathrm{Cof}}{\to }QX\stackrel{\mathrm{Fib}\cap W}{\to }X.$$

So cofibrant replacement comes from the second weak factorization system:

$$(\mathrm{Cof},\mathrm{Fib}\cap W).$$

Fibrant replacement

Dually, consider the canonical morphism

$$X\to \ast .$$

Using the weak factorization system

$$(\mathrm{Cof}\cap W,\mathrm{Fib}),$$

we factor it as

$$X⟶RX⟶\ast$$

where

$$X\to RX$$

is a trivial cofibration, and

$$RX\to \ast$$

is a fibration.

Since

$$RX\to \ast$$

is a fibration, $RX$$RX$ is fibrant.

Since

$$X\to RX$$

is a weak equivalence, $RX$$RX$ is weakly equivalent to $X$$X$.

Thus

$$X\to RX$$

is called a fibrant replacement of $X$$X$.

The diagram is:

$$X\stackrel{\mathrm{Cof}\cap W}{\to }RX\stackrel{\mathrm{Fib}}{\to }\ast .$$

So fibrant replacement comes from the first weak factorization system:

$$(\mathrm{Cof}\cap W,\mathrm{Fib}).$$

Bifibrant replacement

One can combine the two replacements.

First take a cofibrant replacement:

$$QX\to X.$$

Then take a fibrant replacement of $QX$$QX$:

$$QX\to RQX.$$

The object $RQX$$RQX$ is fibrant by construction.

It is also cofibrant. Indeed,

$$QX\to RQX$$

is a trivial cofibration, hence in particular a cofibration. Since

$$\varnothing \to QX$$

is also a cofibration, and cofibrations are closed under composition, the composite

$$\varnothing \to QX\to RQX$$

is a cofibration.

Therefore $RQX$$RQX$ is both cofibrant and fibrant.

Similarly, one may first take a fibrant replacement and then a cofibrant replacement:

$$QRX\to RX.$$

The object $QRX$$QRX$ is cofibrant by construction.

It is also fibrant, because

$$QRX\to RX$$

is a trivial fibration, hence a fibration, and

$$RX\to \ast$$

is a fibration. Since fibrations are closed under composition,

$$QRX\to RX\to \ast$$

is a fibration.

Thus both

$$RQX$$

and

$$QRX$$

are bifibrant replacements of $X$$X$.

Why replacements matter

The main reason to introduce these replacements is that weak equivalences are usually too large and too flexible to compute with directly.

Instead of computing in the localization

$$\mathcal{C}[W^{-1}]$$

by arbitrary zigzags, one replaces objects by better objects.

Cofibrant objects are good sources.

Fibrant objects are good targets.

In a model category, maps in the homotopy category can often be computed by first replacing the source by something cofibrant and the target by something fibrant.

The rough guiding principle is:

$${\mathrm{Hom}}_{\mathcal{C}[W^{-1}]}(X,Y)$$

is computed by maps

$$QX\to RY$$

modulo the appropriate notion of homotopy.

This is why the two weak factorization systems are not decorative. They make localization computable.

Conceptual summary

The structure developed in this post can be summarized as follows.

First, every binary relation gives a contravariant Galois connection.

Second, the lifting relation on morphisms gives the RLP/LLP Galois connection.

Third, the Galois connection gives closed left and right lifting classes.

Fourth, a weak factorization system is a closed lifting pair together with factorizations of all morphisms.

Fifth, a model category is a category with weak equivalences and two compatible weak factorization systems:

$$(\mathrm{Cof}\cap W,\mathrm{Fib})$$

and

$$(\mathrm{Cof},\mathrm{Fib}\cap W).$$

Sixth, these two weak factorization systems produce replacement objects:

$$QX\to X$$

and

$$X\to RX.$$

So the slogan is:

$$\boxed{{\displaystyle \text{Model categories turn localization into replacement calculus.}}}$$

In the next post, we will see this in the category of chain complexes. There, the abstract replacement procedures recover familiar constructions from homological algebra:

$$\text{projective resolutions}$$

and

$$\text{injective resolutions}.$$