science

Reedy category

Reedy category is a science 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 Reedy category rather than just read about it. In short: In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure. A prototypical example is the simplex category or its opposite.

Key takeaways

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

Reference excerpt

In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure. A prototypical example is the simplex category or its opposite. It was introduced by Christopher Reedy in his unpublished manuscript.

Definition A Reedy category consists of the following data: a category R, two wide (lluf) subcategories R − , R + {\displaystyle R_{-},R_{+}} and a functorial factorization of each map into a map in R − {\displaystyle R_{-}} followed by a map in R + {\displaystyle R_{+}} that are subject to the condition: for some total preordering (degree), the nonidentity maps in R − , R + {\displaystyle R_{-},R_{+}} lower or raise degrees. Note some authors such as nlab require each factorization to be unique.

Reedy model structure A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category.

Eilenberg–Zilber category An Eilenberg–Zilber category is a variant of a Reedy category.

References

Literature Barwick, Clark (2007), On Reedy Model Categories, arXiv:0708.2832 Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1-108-47320-0. Clemens Berger, Ieke Moerdijk, On an extension of the notion of Reedy category, Mathematische Zeitschrift, 269, 2011 (arXiv:0809.3341, doi:10.1007/s00209-010-0770-x) Tim Campion, Cubical sites as Eilenberg-Zilber categories, 2023, arXiv:2303.06206

Further reading Reedy category, Reedy model structure and Eilenberg-Zilber category at the nLab http://pantodon.jp/index.rb?body=Reedy_category in Japanese

Worked examples

Example 1 — a first encounter with Reedy category

Start with the simplest possible case. Write down what Reedy category claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Reedy category 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 Reedy category 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 Reedy category

In research: Reedy category appears in science 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 Reedy category 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: Reedy category is common in secondary-school and first-year university syllabi. It links to neighbouring topics Category theory, Category theory stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Reedy category 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Reedy category in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Reedy category 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 Reedy category out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Reedy category in simple terms?

Why does Reedy category matter?

Because it connects several science 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 Reedy category?

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 Reedy category.