# 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.

**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 Mor_{C.}(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 1
_{X}in Mor (X, X) with f1_{X}= f and 1_{X}g 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!

**C.**a category, it is easy to show that the elements required in the second axiom 1

_{X}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 = 1_{X} and fg = 1_{Y} 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 1_{X}(formed in**C.**) in**C '**(X, X).

- for every pair of objects X, Y in O (
**C '**) is**C '**(X, Y) =**C.**(X, Y)

**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,

- 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 (1_{X}) = 1_{F (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:**

- π
_{1}: Top_{*}→ Gr. - H
_{1}: 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, x_{0}) the room X is assigned - one "forgets" the base point; each dotted map f: (X, x_{0}) → (Y, y_{0}) 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 φ_{X}: F (X) → G (X) given,

- For every morphism f: X → Y in
**C.**applies φ_{Y}F (f) = G (f) φ_{X}.- They say: the following diagram in

**D.**is commutative:

**Example of a natural transformation:**

- Look at the functor H
_{1}: Top → Ab as functor Top_{*}→ Gr (i.e. as a series connection of the forgetting functor Top_{*}→ Top, the actual functor H_{1}and the embedding functor Ab → Gr).

The Hurewicz homomorphisms

φ_{(X, x)}: π_{1}(X, x) → H_{1}(X) yield a natural transformation φ: π_{1}→ H_{1}

- What is ather in chemistry
- Can RVs survive an earthquake?
- When will Ripple XRP reach 18
- Is the installation of the fiberglass insulation itself really dangerous?
- How and why did Monkeybox fail
- How have you been articulated so beautifully
- What will be the next acquisition of Facebook
- Why don't I attract men's attention?
- Who created the Salesforce platform
- What is the capital of Mauritania 1
- Why do people choose to follow questions
- Is Bogomolets University good
- Proof of resignation is required in TCS
- What are your healthy coping mechanisms
- Why don't some people like Ford?
- What components create a successful tech startup
- Telephony What was CENTREX
- What is the Bayers Trial
- Bonsai trees are difficult to maintain
- Who invented the concept of computer networks
- Hallucinations are an illusion
- What does isinstance mean in Python
- Like MMORPG, Sword Art Online is
- Did Henry Purcell play an instrument