1 categories

A category C is a directed graph (C_0, C_1) (objects and morphisms) equipped with composition maps

\gamma_{a,b,c} : C(a, b) \times C(b, c) \to C(a, c) : (f, g) \mapsto g \circ f

For \forall_{a, b, c} \in C_0 satisfying the following axioms:

  1. existence of identities - there is a map \text{Id} : C_0 \to C_1 : c \mapsto \text{Id}_c such that \text{Id}_c \in C(c, c) and f \circ \text{Id}_a = f = \text{Id}_b \circ f \forall_{f \in C_1(a, b)} \forall_{a, b \in C_0}
  2. associativity - for any a, b, c, d \in C_0 and f \in C(a, b), g \in C(b, c), and h \in C(c, d) it holds that (h \circ g) \circ f = h \circ (g \circ f) \in C(a, d)

1.1 directed graph (quiver)

A directed graph G consists of a pair of classes G_0 (vertices) and G_1 (edges) equipped with two maps \text{dom} : G_1 \to G_0 and \text{cod} : G_1 \to G_0.

So for a directed graph x \stackrel{f}{\longrightarrow} y, \text{dom}(f) = x and \text{cod}(f) = y

1.1.1 subgraph

A subgraph G' of a directed graph G consists of a pair of subclasses G'_0 \subseteq G_0 and G'_1 \subseteq G_1 such that \text{dom}[G_1'] \cup \text{cod}[G'_1] \subseteq G_0'

1.2 small category

When objects and morphisms are sets then the category is called small.

1.3 subcategory

A subcategory C' of a category C consists of a subgraph (C_0', C_1') that is closed under composition of morphisms and such that \text{Id}_c \in C_1' for all c \in C_0'. It is full if C'(a, b) = C(a, b) for all a, b \in C_0' and is wide if C'_0 = \text{Ob}~C

1.4 examples

  1. \textsf{Set} - objects are all sets and morphisms are all set maps
  2. \textsf{Top} - objects are all topological spaces and morphisms are all continues maps
    1. \textsf{Top}_\star - objects are all based topological spaces and morphisms are all based continues maps
  3. \textsf{Gr} - objects are all groups and morphisms are all group homomorphisms
  4. \textsf{Ab} - objects are all abelian groups and morphisms are all group homomorphisms with abelian domain and codomain
  5. \textsf{Ch}_{\mathbb{k}} - where \mathbb{k} is a commutative ring, objects are all chain complexes of \mathbb{k}-modules and morphisms are all \mathbb{k}-linear chain maps
  6. \textsf{Cat} - objects are all small categories and morphisms are all functors of between small categories

1.5 products

Given two categories C and D, their product is a category C \times D and specified by

\text{Ob}(C \times D) = \text{Ob}~C \times \text{Ob}~D

and

\forall_{c, c' \in \text{Ob}~C}\forall_{d, d' \in \text{Ob}~D} (C \times D)((c, d), (c', d')) = C(c, c') \times D(d, d')

Then \text{Id}_{(c, d)} = (\text{Id}_c, \text{Id}_d)

1.6 category of morphisms

Given a category C we define a category of morphisms C^\to where \text{Ob}~C^\to = \text{Mor}~C and where for every couple f : a \to b, g : c \to d of morphisms of C,

C^\to(f, g) = \{(h : a \to c, k : b \to d) \in \text{Mor}~C \times \text{Mor}~C | g \circ h = k \circ f : a \to d\}

1.7 opposite category

For a category C its opposite category C^{op} is \text{Ob}~C^{op} = \text{Ob}~C and C^{op}(a, b) = C(b, a)

1.8 isomorphism

Given a morphism f : a \to b in C is an isomorphism, denoted \stackrel{\simeq}{\to} if it admits an inverse, ie there is a morphism g : b \to a such that g \circ f = \text{Id}_a and f \circ g = \text{Id}_b. a and b are called isomorphic.

1.9 functors

A morphism between categories. A functor F: C \to D is a pair of maps F_{\text{Ob}} : \text{Ob}~C \to \text{Ob}~D and F_{\text{Mor}} : \text{Mor}~C \to \text{Mor}~D such that:

F_{\text{Mor}}(f) : F_{\text{Ob}}(a) \to F_{\text{Ob}}(b)

where

F_{\text{Mor}}(\text{Id}_c) = \text{Id}_{F_{\text{Ob}(c)}}

and

F_{\text{Mor}}(g \circ f) = F_{\text{Mor}}(g) \circ F_{\text{Mor}}(f)

Functors are composable in an associative manner

1.10 natural transformations

Let F, F' : C \to D be functors. A natural transformation \tau : F \to F' consists of a map

\tau : \text{Ob}~C \to \text{Mor}~D : c \mapsto \tau_c

such that for all c \in \text{Ob}~C

\tau_c \in D(F(c), F'(c))

and all morphisms f : b \to c in C, the diagram commutes

natural transformation diagram

No more notes will appear. I dropped the course due to lack of prerequisite knowledge.