Category theory What is a morphism

Basic categorical concepts 1: categories, functors, natural transformations

The concept of a category

      When one speaks of a "category", one means a class of mathematical objects (which have the same structure) and the structure-preserving mappings between these objects, see the examples below.
One category C. is given by the following data:
  • A class O (C.), the elements in O are called (C.) the Objects the category.
    (Instead of talking about a "class" of objects, one would probably prefer to say that a "set" of objects is given, but because of possible set-theoretical difficulties, one has to access a word other than "set", i.e. the word "Class").
  • For every pair X, Y of objects a set Mor (X, Y), the elements in Mor (X, Y) are called die Morphisms from X to Y and one notices such an element f in Mor (X, Y) also in the form f: X → Y. Instead of Mor (X, Y) one also writes MorC.(X, Y) or also C.(X, Y).
    (The notation f: X → Y is intended to remind us that in most cases the objects are sets with an additional structure, and that the morphisms are set-theoretical maps that respect the additional structure; in general, however, this notation f: X → Y only means, that f belongs to the set Mor (X, Y), and not that "elements" of X are assigned to "elements" of Y - in the context of category theory, objects have no elements!)
  • A link Mor (X, Y) × Mor (Y, Z) → Mor ​​(X, Z); the image of the pair (f, g) with f: X → Y and g: Y → Z under this connection is usually simply noted as gf (or g o f). This link is called the Composition of morphisms.
The following axioms must be fulfilled:
  • (Associativity) If morphisms f: X → Y, g: Y → Z, h: Z → A are given, then (hg) f = h (gf).
  • (Identities) For every object X there is a morphism 1X in Mor (X, X) with f1X = f and 1Xg for and all morphisms f: X → Y and g: W → X.

Examples of categories

We note the objects and the morphisms in each case, actually we should also note the composition of the morphisms, but in all cases it is simply a matter of connecting the images one after the other, or (in the case of the homotopy category of topological spaces) one by connecting in series Illustrations induced surgery. - Of course, we could do without noting the objects, since the name of a category usually simply refers to the objects! Some of the categories are given a name (such as Gr for "groups", Top for "topological spaces") so that they can be used later).
  • The Men category amounts
      OBJECTS: amounts
      MORPHISMS: Illustrations
  • The category size groups
      OBJECTS: groups
      MORPHISMS: Group homorphisms
  • The category from Abelian groups
      OBJECTS: Abelian groups
      MORPHISMS: Group homorphisms
  • The category of Vector spaces (over a solid body k)
      OBJECTS: k-vector spaces
      MORPHISMS: k-linear mappings
  • The category of Rings
      OBJECTS: Rings
      MORPHISMS: Ring homorphisms
  • The top category of topological spaces
      OBJECTS: topological spaces
      MORPHISMS: continuous mappings
  • The top category* the dotted topological spaces
      OBJECTS: dotted topological spaces
      MORPHISMS: continuous mappings that map base point to base point
  • The category of metric spaces
      OBJECTS: metric spaces
      MORPHISMS: continuous mappings
  • The Homotopy category of topological spaces
      OBJECTS: topological spaces
      MORPHISMS: Homotopy classes of continuous mappings
      In contrast to all previous examples, here are the morphisms no Images, but equivalence classes of images!
Is C. a category, it is easy to show that the elements required in the second axiom 1X are uniquely determined in each case (as one shows with groups that there is only one elementary element).

Are morphisms f: X → Y and g: Y → X with gf = 1X and fg = 1Y given, we call f, g (inverse to each other) Isomorphisms and one then says that the objects X, Y isomorphic are.

It should be clear what is meant by a sub-category: C ' is a Subcategory the category C.if:

  1. O(C ') is a subclass of O (C.).
  2. for every pair of objects X, Y in O (C ') is C '(X, Y) a subset of C.(X, Y),
  3. Is f in C '(X, Y) and g in C '(Y, Z), so belongs gf (formed in C.) to C '(X, Z).
  4. For every X in C ' is 1X (formed in C.) in C '(X, X).
Even applies in (2):
  • for every pair of objects X, Y in O (C ') is C '(X, Y) = C.(X, Y)
that's what they call C ' a full Subcategory of C.. In this case, conditions (3) and (4) are automatically satisfied.

Example: The Abelian Groups category is a full sub-category of the Gr category of Groups.

Functors

Be C. and D. Categories. A functor F:C.D. is given by the following data:
  • Every object X in O (C.) is an object F (X) in O (D.) assigned.
  • Every morphism f in C.(X, Y) is a morphism F (f) in D.(F (X), F (Y)) assigned,
where the following two conditions must be met:
  • Are morphisms f: X → Y and g: Y → Z in C. given, then F (gf) = F (g) F (f).
  • For each object X in C. is F (1X) = 1F (X).
    The functors defined in this way are also called covariant functors, in contrast to the "contravariant functors" (in which every morphism f in C.(X, Y) a morphism F (f) in D.(F (Y), F (X)) (and not in D.(F (X), F (Y))) and for which F (gf) = F (f) F (g) (instead of F (gf) = F (g) F (f)) is required).

Examples of functors:

  • π1: Top* → Gr.
  • H1: Top → Ab.
  • Is C ' a sub-category of the category C., so inclusion is a functor, you call it one Embedding function.
    • Example: the embedding function Ab → Gr.
  • Forget functors: These are functors that are given by forgetting a given additional structure.
      Examples:
    • The forget function Gr → Meng (each group is assigned the underlying set, each group homomorphism f is assigned this mapping f, now simply expressed as a set-theoretic mapping).
    • The forgetting functor Top → Meng (each topological space is assigned the underlying set, each continuous mapping f this mapping f, now simply expressed as a set-theoretic mapping).
    • The forgetful functor top* → Top (every dotted space (X, x0) the room X is assigned - one "forgets" the base point; each dotted map f: (X, x0) → (Y, y0) this mapping f, now simply interpreted as mapping f: X → Y, is assigned).

Natural transformations

Be C. and D. Categories. Let F, G:C.D. Functors. A natural transformation φ: F → G is given as follows:
  • For every object X in O (C.) is a morphism φX: F (X) → G (X) given,
so that:
  • For every morphism f: X → Y in C. applies φYF (f) = G (f) φX.
      They say: the following diagram in D. is commutative:

Example of a natural transformation:

  • Look at the functor H1: Top → Ab as functor Top* → Gr (i.e. as a series connection of the forgetting functor Top* → Top, the actual functor H1 and the embedding functor Ab → Gr).
    The Hurewicz homomorphisms
    φ(X, x): π1(X, x) → H1(X) yield a natural transformation φ: π1 → H1