Ken Brown Lemma

Ken Brown's Lemma is a fundamental result about when functors preserve weak equivalences. It provides the main tool for constructing derived functors and establishing that functors descend to the homotopy category.

TheoremKen Brown's Lemma

Let $F: \mathcal{M} \to \mathcal{N}$ be a functor between model categories. If $F$ preserves fibrations (resp. cofibrations) and trivial fibrations (resp. trivial cofibrations), then $F$ preserves weak equivalences between fibrant (resp. cofibrant) objects.

RemarkWhy This Is Important

The lemma says we don't need to verify directly that a functor preserves all weak equivalences—it suffices to check the technically simpler condition of preserving (trivial) fibrations or (trivial) cofibrations. This is crucial because:

Fibrations and cofibrations are often characterized by explicit lifting properties
Weak equivalences can be difficult to detect directly
The lemma converts a homotopy-theoretic condition into a categorical one

ExampleDerived Tensor Product

For the tensor product functor $- \otimes Y: \mathbf{Ch}(\mathbb{Z}) \to \mathbf{Ch}(\mathbb{Z})$ , Ken Brown's Lemma shows:

If $Y$ is cofibrant (projective), then $X \otimes Y$ preserves weak equivalences between cofibrant objects. This justifies defining the derived tensor product as: $X \otimes^{\mathbf{L}} Y = P(X) \otimes P(Y)$ where $P$ denotes projective replacement.

ProofSketch of Proof

Suppose $F$ preserves fibrations and trivial fibrations. Let $f: X \to Y$ be a weak equivalence between fibrant objects. By the factorization axiom, factor $f$ as: $X \xrightarrow{j} Z \xrightarrow{q} Y$ where $j$ is a trivial cofibration and $q$ is a fibration.

By two-out-of-three, $q$ is also a weak equivalence, hence a trivial fibration. Since $F$ preserves trivial fibrations, $F(q)$ is a trivial fibration, hence a weak equivalence.

The key observation is that $F(j)$ can be shown to be a weak equivalence using the cylinder construction and the fact that $F$ preserves trivial fibrations. Then by two-out-of-three: $F(f) = F(q) \circ F(j)$ is a weak equivalence.

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DefinitionRight/Left Quillen Functors

A functor $F: \mathcal{M} \to \mathcal{N}$ is:

Right Quillen if it has a left adjoint and preserves fibrations and trivial fibrations
Left Quillen if it has a right adjoint and preserves cofibrations and trivial cofibrations

Ken Brown's Lemma immediately implies that right Quillen functors preserve weak equivalences between fibrant objects, and left Quillen functors preserve weak equivalences between cofibrant objects.

ExampleGeometric Realization

The geometric realization functor $|-|: \mathbf{sSet} \to \mathbf{Top}$ is left Quillen because it:

Preserves cofibrations (monomorphisms map to cofibrations)
Preserves trivial cofibrations

Ken Brown's Lemma then guarantees that $|-|$ preserves weak equivalences between cofibrant objects. Since all simplicial sets are cofibrant, $|-|$ preserves all weak equivalences.

RemarkApplications to Derived Functors

Ken Brown's Lemma is the main tool for showing that derived functors are well-defined. For a left Quillen functor $F$ , the derived functor: $\mathbf{L}F: \text{Ho}(\mathcal{M}) \to \text{Ho}(\mathcal{N})$ $\mathbf{L}F(X) = F(QX)$ is well-defined because Ken Brown's Lemma ensures that different cofibrant replacements $QX$ give isomorphic results in $\text{Ho}(\mathcal{N})$ .

ExampleHom Functors

For a fixed cofibrant object $A$ in a simplicial model category, the functor $\text{Map}(A, -): \mathcal{M} \to \mathbf{sSet}$ preserves fibrations and trivial fibrations.

Ken Brown's Lemma implies it preserves weak equivalences between fibrant objects, making it a well-defined functor on the homotopy category.

Ken Brown's Lemma elegantly bridges the gap between the technical machinery of model categories (fibrations and cofibrations) and the homotopy-theoretic goal (preserving weak equivalences), making it indispensable for working with derived functors.