Quotient category

Key takeaways

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

Reference excerpt

In mathematics, a quotient category is a category obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in the categorical setting.

Definition Let C {\displaystyle \mathbf {C} } be a category. A congruence relation R {\displaystyle {\mathcal {R}}} on C {\displaystyle \mathbf {C} } is given by: for each pair of objects X {\displaystyle X} , Y {\displaystyle Y} in C {\displaystyle \mathbf {C} } , an equivalence relation R X , Y {\displaystyle {\mathcal {R}}_{X,Y}} on H o m ( X , Y ) {\displaystyle \mathrm {Hom} (X,Y)} , such that the equivalence relations respect composition of morphisms. That is, if

f 1 , f 2 : X → Y {\displaystyle f_{1},f_{2}:X\to Y\,}

are related in H o m ( X , Y ) {\displaystyle \mathrm {Hom} (X,Y)} and

g 1 , g 2 : Y → Z {\displaystyle g_{1},g_{2}:Y\to Z\,}

are related in H o m ( Y , Z ) {\displaystyle \mathrm {Hom} (Y,Z)} , then g 1 f 1 {\displaystyle g_{1}f_{1}} and g 2 f 2 {\displaystyle g_{2}f_{2}} are related in H o m ( X , Z ) {\displaystyle \mathrm {Hom} (X,Z)} . Given a congruence relation R {\displaystyle {\mathcal {R}}} on C {\displaystyle \mathbf {C} } we can define the quotient category C / R {\displaystyle \mathbf {C} /{\mathcal {R}}} as the category whose objects are those of C {\displaystyle \mathbf {C} } and whose morphisms are equivalence classes of morphisms in C {\displaystyle \mathbf {C} } . That is,

H o m C / R ( X , Y ) = H o m C ( X , Y ) / R X , Y . {\displaystyle \mathrm {Hom} _{\mathbf {C} /{\mathcal {R}}}(X,Y)=\mathrm {Hom} _{\mathbf {C} }(X,Y)/{\mathcal {R}}_{X,Y}.}

Composition of morphisms in C / R {\displaystyle \mathbf {C} /{\mathcal {R}}} is well-defined since R {\displaystyle {\mathcal {R}}} is a congruence relation.

Properties There is a natural quotient functor from C {\displaystyle \mathbf {C} } to C / R {\displaystyle \mathbf {C} /{\mathcal {R}}} which sends each morphism to its equivalence class. This functor is bijective on objects and surjective on Hom-sets (i.e. it is a full functor). Every functor F : C → D {\displaystyle F\colon \mathbf {C} \to \mathbf {D} } determines a congruence on C {\displaystyle \mathbf {C} } by saying f ∼ g {\displaystyle f\sim g} iff F ( f ) = F ( g ) {\displaystyle F(f)=F(g)} . The functor F {\displaystyle F} then factors through the quotient functor C → C / ∼ {\displaystyle \mathbf {C} \to \mathbf {C} /\sim } in a unique manner. This may be regarded as the "first isomorphism theorem" for categories.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Quotient category

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

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

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

Frequently asked questions

What is Quotient category in simple terms?

In mathematics, a quotient category is a category obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in the categorical setting.

Why does Quotient 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 Quotient 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 Quotient category.