science

Simplicially enriched category

Simplicially enriched 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 Simplicially enriched category rather than just read about it. In short: In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often also called, more ambiguously, simplicial categories; the latter term however also applies to simplicial objects in Cat (the category of small categories).

Key takeaways

Simplicially enriched 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 Simplicially enriched category to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Simplicially enriched category from memory before moving on to harder problems.

Reference excerpt

In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often also called, more ambiguously, simplicial categories; the latter term however also applies to simplicial objects in Cat (the category of small categories). Simplicially enriched categories can, however, be identified with simplicial objects in Cat whose object part is constant, or more precisely, all face and degeneracy maps are bijective on objects. Simplicially enriched categories can model (∞, 1)-categories, but the dictionary has to be carefully built. Namely, many notions (limits, for example) are different from the limits in the sense of enriched category theory. The homotopy coherent nerve of a simplicially enriched category is a simplicial set that generalizes the nerve of a category.

References Goerss, Paul; Jardine, John (2009), Simplicial Homotopy Theory, Progress in Mathematics, vol. 174, Birkhäuser Basel, ISBN 978-3-7643-6064-1, MR 1711612 Lurie, Jacob (2009), Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, arXiv:math.CT/0608040, ISBN 978-0-691-14049-0, MR 2522659

External links Simplicially enriched category at the nLab

Worked examples

Example 1 — a first encounter with Simplicially enriched category

Start with the simplest possible case. Write down what Simplicially enriched 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 Simplicially enriched 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 Simplicially enriched 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 Simplicially enriched category

In research: Simplicially enriched 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 Simplicially enriched 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: Simplicially enriched category is common in secondary-school and first-year university syllabi. It links to neighbouring topics Higher category theory, Simplicial sets, so understanding it makes those chapters shorter.
In everyday life: Look for Simplicially enriched 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.

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How to study Simplicially enriched category in 20 minutes

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

Frequently asked questions

What is Simplicially enriched category in simple terms?

Why does Simplicially enriched 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 Simplicially enriched 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 Simplicially enriched category.