Homotopical algebra start

.vscode/settings.json

@@ -54,6 +54,7 @@
     "CMAC",
     "CMYK",
     "Cocke",
+    "codomain",
     "cofactor",
     "combinatorics",
     "concretization",
@@ -178,6 +179,8 @@
     "Mobius",
     "monic",
     "Monika",
+    "morphism",
+    "morphisms",
     "MPCP",
     "MRMW",
     "MRSW",
@@ -298,6 +301,7 @@
     "Tarapata",
     "Tarski",
     "textit",
+    "textsf",
     "tka",
     "Tomasz",
     "totient",

masters/homotopical_algebra/categories.md

@@ -0,0 +1,71 @@
+# 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 $(h \circ g) \circ f = h \circ (g \circ f) \in C(a, d)$
+
+## 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 $\text{cod}(f) = y$
+
+### 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 $\text{dom}[G_1'] \cup \text{cod}[G'_1] \subseteq G_0'$
+
+## small category
+
+When objects and morphisms are sets then the category is called small.
+
+## 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 $C'_0 = \text{Ob}~C$
+
+## 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
+
+## 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 $\text{Id}_{(c, d)} = (\text{Id}_c, \text{Id}_d)$
+
+## 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 $C$,
+
+$$
+C^\to(f, g) = \{(h : a \to c, k : b \to d) \in \text{Mor}~C \times \text{Mor}~C | g \circ h = k \circ f : a \to d\}
+$$
+
+## opposite category
+
+For a category $C$ its opposite category $C^{op}$ is $\text{Ob}~C^{op} = \text{Ob}~C$ and $C^{op}(a, b) = C(b, a)$
+
+## 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.