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.
- 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.
- (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 categoriesWe 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!
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:
- O(C ') is a subclass of O (C.).
- for every pair of objects X, Y in O (C ') is C '(X, Y) a subset of C.(X, Y),
- Is f in C '(X, Y) and g in C '(Y, Z), so belongs gf (formed in C.) to C '(X, Z).
- For every X in C ' is 1X (formed in C.) in C '(X, X).
- for every pair of objects X, Y in O (C ') is C '(X, Y) = C.(X, Y)
Example: The Abelian Groups category is a full sub-category of the Gr category of Groups.
FunctorsBe 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,
- 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.
- 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 transformationsBe 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,
- 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
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