science

Functor

Functor 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 Functor rather than just read about it. In short: In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces.

Key takeaways

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

Reference excerpt

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in every area of mathematics where category theory is applied. The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used functor in a linguistic context; see function word.

Definition

Let C {\displaystyle C} and D {\displaystyle D} be categories. A functor F {\displaystyle F} from C {\displaystyle C} to D {\displaystyle D} is a mapping that:

associates each object X {\displaystyle X} in C {\displaystyle C} to an object F ( X ) {\displaystyle F(X)} in D {\displaystyle D} , associates each morphism f : X → Y {\displaystyle f:X\to Y} in C {\displaystyle C} to a morphism F ( f ) : F ( X ) → F ( Y ) {\displaystyle F(f):F(X)\to F(Y)} in D {\displaystyle D} such that the following two conditions hold:

F ( i d X ) = i d F ( X ) {\displaystyle F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!} for every object X {\displaystyle X} in C {\displaystyle C} ,

F ( g ∘ f ) = F ( g ) ∘ F ( f ) {\displaystyle F(g\circ f)=F(g)\circ F(f)} for all morphisms f : X → Y {\displaystyle f:X\to Y\,\!} and g : Y → Z {\displaystyle g:Y\to Z} in C {\displaystyle C} . That is, functors must preserve identity morphisms and composition of morphisms.

Covariance and contravariance

There are many constructions in mathematics that would be functors, except that they "turn morphisms around" and "reverse composition". We then define a contravariant functor F {\displaystyle F} from C {\displaystyle C} to D {\displaystyle D} as a mapping that

associates each object X {\displaystyle X} in C {\displaystyle C} with an object F ( X ) {\displaystyle F(X)} in D {\displaystyle D} , associates each morphism f : X → Y {\displaystyle f:X\to Y} in C {\displaystyle C} with a morphism F ( f ) : F ( Y ) → F ( X ) {\displaystyle F(f):F(Y)\to F(X)} in D {\displaystyle D} such that the following two conditions hold:

F ( i d X ) = i d F ( X ) {\displaystyle F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!} for every object X {\displaystyle X} in C {\displaystyle C} ,

F ( g ∘ f ) = F ( f ) ∘ F ( g ) {\displaystyle F(g\circ f)=F(f)\circ F(g)} for all morphisms f : X → Y {\displaystyle f:X\to Y} and g : Y → Z {\displaystyle g:Y\to Z} in C {\displaystyle C} . Variance of functors (composite):

The composite of two functors of the same variance:

C o v a r i a n t ∘ C o v a r i a n t → C o v a r i a n t {\displaystyle \mathrm {Covariant} \circ \mathrm {Covariant} \to \mathrm {Covariant} }

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Functor

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

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

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

Frequently asked questions

What is Functor in simple terms?

Why does Functor 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 Functor?

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 Functor.