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I have numerous questions on my abstract homework asking me to define "the natural map", though i don't see reference to it in my textbook.

Let X and Y be sets and let C be the set {f : {1,2}→ X ∪Y|f(1) ∈ X and f(2) ∈ Y} (1) Define the natural map Γ: C → X × Y (2) Define the map $Γ^{−1}$: X ×Y → C .

I don't understand what is being asked of me. Also:

For a set S define the natural isomorphism Char: P(S) →F(S,{0,1}). For A ⊆ S denote the function Char(A) by χA. (2) Define Char−1: F(S,0,1) →P(S)

I know that an isomorphism is a function that perserves structure between two structurally-the-same algebraic thingies.

Can someone give me an intuitive definition of what all this natural stuff is about?

Mathemagician1234
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user121615
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    Technically the precise definition of naturality is as in category theory. A working definition for things not in this context is that a natural map $\varphi:X\rightarrow Y$ doesn't depend on the specificities of $X$ and $Y$. For example, when $X$ and $Y$ are vector spaces, this would be (in some senses) tantamount to not picking particular bases to define the map. – Hayden Feb 16 '15 at 23:21
  • So, basically, what you are saying is that the natural map doesn't really depend on what element I pick for x or Y? – user121615 Feb 16 '15 at 23:24
  • I prefer to use "canonical" for the way the author is using "natural" here (there's only one obvious guess), and reserve "natural" for when maps induce commutative diagrams. – DCT Feb 16 '15 at 23:49

4 Answers4

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Natural maps arise all the time in algebra and topology and it's important to understand the definition. A canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. A closely related notion is a structure map or structure morphism; the map that comes with the given structure on the object. They are also sometimes called canonical maps.Canonical maps are usually associated with quotient structures, which allow one to generalize the idea of congruence to abstract algebriac objects or topological ones, such as quotient spaces.

The best way to understand what a natural or canonical map is to see some examples.

1) If N is a normal subgroup of a group G, then there is a canonical map from G to the quotient group G/N that sends an element g to the coset that g belongs to. 2) If V is a vector space, then there is a canonical map from V to the second dual space of V that sends a vector v to the linear functional fv defined by fv(λ) = λ(v). 3) If f is a ring homomorphism from a commutative ring R to commutative ring S, then S can be viewed as an algebra over R. The ring homomorphism f is then called the structure map (for the algebra structure). The corresponding map on the prime spectra: Spec(S) →Spec(R) is also called the structure map. 4) If E is a vector bundle over a topological space X, then the projection map from E to X is the structure map.

In this case, the natural map is the indexed ordered pair map of $X\cup Y$ into X x Y . Clearly, the map f that defines C is the indexing map of the ordered pairs in X x Y where $x_1\in X$ and $x_2\in Y$. So that Γ: C → X × Y is defined by Γ($x_1,x_2$) = ($x_1,x_2$) where $x_1,x_2$ is the unordered indexed pair in C and ($x_1,x_2$) an ordered pair in X x Y.

Can you now define the inverse? First,you have to show this map is an isomorphism, which isn't hard.

Mathemagician1234
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A natural map is a map which you are too lazy to define.

Nicolas Bourbaki
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  • It has nothing to do with laziness. I just don't understand what it means. – user121615 Feb 16 '15 at 23:29
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    @user I was not talking about you! I was saying that, when an author is too lazy to write out explicitly a formula for an "obvious" map, he just calls it the "natural" map. In mathematics it is important not to write every single detail. The use of the word "natural", is one such example. – Nicolas Bourbaki Feb 16 '15 at 23:30
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    omg I'm so sorry. haha. I feel like an idiot. – user121615 Feb 16 '15 at 23:31
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    @user Do not worry. Every mathematician, or an aspiring mathematician, is a friend of mine. – Nicolas Bourbaki Feb 16 '15 at 23:32
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    Honestly, having just stumbled upon this: this is the best, if not the most helpful, answer. +1 – The Count May 30 '17 at 14:18
  • @NicolasBourbaki,the best answer ever. Does "structure map" have the same meaning that you have mentioned for natural map? – Nil Feb 19 '21 at 02:50
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Intuitively speaking, to say that a map is "natural" usually just means it's defined independently of any choices, so that if you and I are given the same data, we will come up with the same map.

More rigorously, most things that are called "natural maps" can be formulated in the language of category theory as natural transformations between functors.

Jack Lee
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In this context "natural" means "the simplest, least surprising one". The image of $f$ is a pair $\{x,y\}$ such that $x\in X$ and $y\in Y$. The natural value for $\Gamma(f)$ would be $(x,y)$.

vadim123
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