mathematics

Quillen adjunction

Quillen adjunction is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Quillen adjunction rather than just read about it. In short: In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen.

Key takeaways

Quillen adjunction belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Quillen adjunction to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Quillen adjunction from memory before moving on to harder problems.

Reference excerpt

In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen.

Formal definition Given two closed model categories C and D, a Quillen adjunction is a pair

(F, G): C ⇆ {\displaystyle \leftrightarrows } D of adjoint functors with F left adjoint to G such that F preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that G preserves fibrations and trivial fibrations. In such an adjunction F is called the left Quillen functor and G is called the right Quillen functor.

Properties It is a consequence of the axioms that a left (right) Quillen functor preserves weak equivalences between cofibrant (fibrant) objects. The total derived functor theorem of Quillen says that the total left derived functor

LF: Ho(C) → Ho(D) is a left adjoint to the total right derived functor

RG: Ho(D) → Ho(C). This adjunction (LF, RG) is called the derived adjunction. If (F, G) is a Quillen adjunction as above such that

F(c) → d with c cofibrant and d fibrant is a weak equivalence in D if and only if

c → G(d) is a weak equivalence in C then it is called a Quillen equivalence of the closed model categories C and D. In this case the derived adjunction is an adjoint equivalence of categories so that

LF(c) → d is an isomorphism in Ho(D) if and only if

c → RG(d) is an isomorphism in Ho(C).

References Goerss, Paul G. [in German]; Jardine, John F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. Vol. 174. Basel, Boston, Berlin: Birkhäuser. ISBN 978-3-7643-6064-1. [1] [2] Philip S. Hirschhorn, Model Categories and Their Localizations, American Mathematical Soc., Aug 24, 2009 - Mathematics - 457 pages

Worked examples

Example 1 — a first encounter with Quillen adjunction

Start with the simplest possible case. Write down what Quillen adjunction claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Quillen adjunction before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Quillen adjunction ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Quillen adjunction

In research: Quillen adjunction appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Quillen adjunction in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Quillen adjunction is common in secondary-school and first-year university syllabi. It links to neighbouring topics Adjoint functors, Homotopy theory, Theory of continuous functions, so understanding it makes those chapters shorter.
In everyday life: Look for Quillen adjunction outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Quillen adjunction in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quillen adjunction means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Quillen adjunction out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quillen adjunction in simple terms?

Why does Quillen adjunction matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Quillen adjunction?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Quillen adjunction.