In mathematics, specifically in homotopy theory, Ken Brown's lemma gives a sufficient condition for a functor on a category of fibrant objects to preserve weak equivalences; the sufficient condition is that acyclic fibrations go to weak equivalences. The dual of the statement also holds. The lemma or, more precisely, a result of which the lemma is a corollary, was introduced by Kenneth Brown.
Proof The lemma follows from the following:
To see the lemma follows from the above, let w {\displaystyle w} be a weak equivalence and F {\displaystyle F} the given functor. By the factorization lemma, we can write
w = g ∘ j {\displaystyle w=g\circ j}
with an acyclic fibration r {\displaystyle r} such that r ∘ j = id {\displaystyle r\circ j=\operatorname {id} } . Note j {\displaystyle j} is a weak equivalence since r {\displaystyle r} is. Thus, g {\displaystyle g} is a weak equivalence (thus acyclic fibration) since w {\displaystyle w} is. So, F ( g ) {\displaystyle F(g)} is a weak equivalence by assumption. Similarly, F ( j ) {\displaystyle F(j)} is a weak equivalence. Hence, F ( w ) = F ( g ) ∘ F ( j ) {\displaystyle F(w)=F(g)\circ F(j)} is a weak equivalence. ◻ {\displaystyle \square }
Proof of factorization lemma: Let f : X → Y {\displaystyle f:X\to Y} be the given morphism. Let
e : Y I → Y × Y {\displaystyle e:Y^{I}\to Y\times Y}
be the path object fibration; namely, it is obtained by factorizing the diagonal map Δ {\displaystyle \Delta } as Δ = e ∘ c {\displaystyle \Delta =e\circ c} where c {\displaystyle c} is a weak equivalence. Then let π : Z → X × Y {\displaystyle \pi :Z\to X\times Y} be the pull-back of e {\displaystyle e} along f × id {\displaystyle f\times \operatorname {id} } , which is again a fibration. Then by the universal property of the pull-back, we get a map j : X → Z {\displaystyle j:X\to Z} so that the resulting diagram with X → f Y → c Y I {\displaystyle X{\overset {f}{\to }}Y{\overset {c}{\to }}Y^{I}} and ( id X , f ) {\displaystyle (\operatorname {id} _{X},f)} commutes. Take g {\displaystyle g} to be Z → π X × Y → p 2 Y {\displaystyle Z{\overset {\pi }{\to }}X\times Y{\overset {p_{2}}{\to }}Y} , which is a fibration since the projection p 2 {\displaystyle p_{2}} is the pull-back of the fibration Y → {\displaystyle Y\to } final object. As for j {\displaystyle j} , let r {\displaystyle r} be Z → π X × Y → p 1 X {\displaystyle Z{\overset {\pi }{\to }}X\times Y{\overset {p_{1}}{\to }}X} , which is again a fibration. Note that r {\displaystyle r} is the pull-back of q ∘ e : Y I → Y {\displaystyle q\circ e:Y^{I}\to Y} , q {\displaystyle q} a projection. Since e ∘ c = Δ {\displaystyle e\circ c=\Delta } , we have q ∘ e ∘ c = id {\displaystyle q\circ e\circ c=\operatorname {id} } . It follows q ∘ e {\displaystyle q\circ e} is a weak equivalence (since c {\displaystyle c} is) and thus r {\displaystyle r} is a weak equivalence. ◻ {\displaystyle \square }
References
Cisinski, Denis-Charles (2019-06-30). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200. Joyal, André; Tierney, Myles (2008). "Notes on simplicial homotopy theory" (PDF). "factorization lemma". ncatlab.org.
Further reading https://math.stackexchange.com/questions/4721727/understanding-a-proof-of-ken-browns-lemma https://mathoverflow.net/questions/342085/where-is-browns-lemma-from