Internal category

Key takeaways

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

Reference excerpt

In mathematics, more specifically in category theory, internal categories are a generalization of the notion of a small category, and are defined with respect to a fixed ambient category. If the ambient category is taken to be the category of sets then one recovers the theory of small categories. In general, internal categories consist of a pair of objects in the ambient category—thought of as the 'object of objects' and 'object of morphisms'—together with a collection of morphisms in the ambient category satisfying certain identities. Group objects are common examples of internal categories. There are notions of internal functors and natural transformations that make the collection of internal categories in a fixed category into a 2-category.

Definitions Let C {\displaystyle C} be a category with pullbacks. An internal category in C {\displaystyle C} consists of the following data: two C {\displaystyle C} -objects C 0 , C 1 {\displaystyle C_{0},C_{1}} named "object of objects" and "object of morphisms" respectively and four C {\displaystyle C} -arrows d 0 , d 1 : C 1 → C 0 , e : C 0 → C 1 , m : C 1 × C 0 C 1 → C 1 {\displaystyle d_{0},d_{1}:C_{1}\rightarrow C_{0},e:C_{0}\rightarrow C_{1},m:C_{1}\times _{C_{0}}C_{1}\rightarrow C_{1}} subject to coherence conditions expressing the axioms of category theory. See

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See also Enriched category Double category

References

Internal category at the nLab

Worked examples

Example 1 — a first encounter with Internal category

Start with the simplest possible case. Write down what Internal 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 Internal 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 Internal 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 Internal category

In research: Internal 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 Internal 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: Internal 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 Internal 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.

How to study Internal category in 20 minutes

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

Frequently asked questions

What is Internal category in simple terms?

In mathematics, more specifically in category theory, internal categories are a generalization of the notion of a small category, and are defined with respect to a fixed ambient category. If the ambient category is taken to be the category of sets then one recovers the theory of small categories.

Why does Internal 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 Internal 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 Internal category.