mathematics

Semigroupoid

Semigroupoid 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 Semigroupoid rather than just read about it. In short: In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a small category, except possibly for the requirement that there be an identity at each object. While this definition is due to Tilson, Exel has introduced a different definition, one in which there is no underlying graph.

Key takeaways

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

Reference excerpt

In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a small category, except possibly for the requirement that there be an identity at each object. While this definition is due to Tilson, Exel has introduced a different definition, one in which there is no underlying graph. The term semicategory usually refers to a Tilson's graphed semigroupoid. Semigroupoids generalise semigroups in the same way that small categories generalise monoids and groupoids generalise groups. Semigroupoids have applications in the structural theory of semigroups. Formally, a semigroupoid consists of:

a set of things called objects. for every two objects A and B a set Mor(A,B) of things called morphisms from A to B. If f is in Mor(A,B), we write f : A → B. for every three objects A, B and C a binary operation Mor(A,B) × Mor(B,C) → Mor(A,C) called composition of morphisms. The composition of f : A → B and g : B → C is written as g ∘ f or gf. (Some authors write it as fg.) such that the following axiom holds:

(associativity) if f : A → B, g : B → C and h : C → D then h ∘ (g ∘ f) = (h ∘ g) ∘ f.

Examples The Yoneda lemma does not hold in general for semicategories.

References

Mitchell, Barry (1972). "The Dominion of Isbell". Transactions of the American Mathematical Society. 167: 319–331. doi:10.1090/S0002-9947-1972-0294441-0. JSTOR 1996142. Moens, M.; Berni-Canani, U.; Borceux, F. (2002). "On regular presheaves and regular semi-categories" (PDF). Cahiers de Topologie et Géométrie Différentielle Catégoriques. Stubbe, Isar (2005). "Categorical structures enriched in a quantaloid : regular presheaves, regular semicategories" (PDF). Cahiers de Topologie et Géométrie Différentielle Catégoriques. 46 (2): 99–121.

External links "Yoneda lemma 6. The Yoneda lemma in semicategories". ncatlab.org. The Univalent Foundations Program (2013). "Homotopy Type Theory: Univalent Foundations of Mathematics". Homotopy Type Theory.

Worked examples

Example 1 — a first encounter with Semigroupoid

Start with the simplest possible case. Write down what Semigroupoid 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 Semigroupoid 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 Semigroupoid 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 Semigroupoid

In research: Semigroupoid 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 Semigroupoid 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: Semigroupoid is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic structures, Category theory, Category theory stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Semigroupoid 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 Semigroupoid in 20 minutes

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

Frequently asked questions

What is Semigroupoid in simple terms?

Why does Semigroupoid 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 Semigroupoid?

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 Semigroupoid.