Quantum Homotopy Type Theory

Definition 1: Binary Operation A binary operation on a set \(A\) is a function \(A^2 \rightarrow A\). Definition 2: Field A field is a triple \((K, +, \cdot)\) where \(K\) is a set and \(+\) and \(\cdot\) are binary operations on \(K\), called addition and multiplication , such that: (1) \((x + y) + z = x + (y + z)\) and \((xy)z = x(yz)\) for all \(x,y,z \in \mathbb{R}\) (2) \(x+y=y+x\) and \(xy=yx\) for all \(x,y \in \mathbb{R}\) (3) There exists an element \(0 \in \mathbb{R}\), called zero , such that \(x + 0 = x\) for all \(x \in \mathbb{R}\). There exists an element \(1 \in \mathbb{R}\), called one , such that \(0 \neq 1\) and \(x1=x\) for all \(x \in \mathbb{R}\). (4) For all \(x \in \mathbb{R}\), there exists a unique \(y \in \mathbb{R}\) such that \(x + y = 0\), called the negative of \(x\) and denoted \(-x\). For each \(x \in \mathbb{R}\) different from 0, there exists a unique \(y \in \mathbb{R}\) such that \(xy = 1\), called the reciprocal of \(x...

Definition 1: Relation A relation on a set \(A\) is a subset of \(A^2\). Given a relation \(R\), we write \(xRy\) for \((x,y) \in R\). Equivalence Relations and Partitions Definition 2: Reflexive A relation \(R\) on \(A\) is reflexive iff \( \forall x \in A. xRx \). Definition 3: Symmetric A relation \(R\) on \(A\) is symmetric iff \( \forall x,y \in A. x R y \Rightarrow y R x \). Definition 4: Transitive A relation \(R\) on \(A\) is transitive iff \( \forall x,y,z \in A. xRy \wedge yRz \Rightarrow xRz \). Definition 5: Equivalence Relation An equivalence relation on a set \(A\) is a relation on \(A\) that is reflexive, symmetric and transitive. A side remark here. Munkres gives as an example the relation of 'blood relative', defined as \(x\) is a blood relative of \(y\) iff \(x\) and \(y\) have a common ancestor, and then says this relation is not transitive - "I am not a blood relative of my wife, although my children are". This is no...

Definition 1: Function A function is a triple \((A,B,f)\) where \(A\) and \(B\) are sets and \(f \subseteq A \times B\) is such that, for all \(x \in A\), there exists a unique \(y \in B\) such that \((x,y) \in f\). We often write just \(f\) for \((A,B,f)\). We call \(A\) the domain of \(f\) and \(B\) the codomain of \(f\). We write \(f : A \rightarrow B\) to mean \(f\) is a function with domain \(A\) and codomain \(B\). The range or image of \(f\) is the set \(\{y \in B \mid \exists x \in A. (x,y) \in f \}\). For \(x \in A\), we write \(f(x)\) for the unique \(y\) such that \((x,y) \in f\), and call this the value of \(f\) at \(x\). Definition 2: Restriction Let \(f : A \rightarrow B\) and \(C \subseteq A\). The restriction of \(f\) to \(C\) is the function \(f \restriction C : C \rightarrow B \) defined by \((f \restriction C)(x) = f(x)\) for \(x \in C\). Definition 3: Composition Given functions \(f : A \rightarrow B\) and \(g : B \rightarrow C\), the composi...

Munkres uses some unusual notation in his book - in particular, he uses \(a \times b\) for the ordered pair \((a,b)\). I'm going to be using my own preferred notation throughout these notes, to make things consistent between books. The book begins with a recap of naive set theory, going into quite some depth (up to the Well-Ordering Theorem). So let's begin at the beginning: Basic Notation We write \(a \in A\) iff \(A\) is a set and \(a\) is a member of \(A\), and \(a \notin A\) iff \(a\) is not a member of \(A\). Two sets are equal iff they have exactly the same elements. We say \(A\) is a subset of \(B\), or \(B\) includes \(A\), and write \(A \subseteq B\), iff every element of \(A\) is an element of \(B\). We say \(A\) is a proper subset of \(B\), or \(B\) properly includes \(A\), and write \(A \subsetneq B\), iff in addition \(A \neq B\). We write \(\{a,b,c\}\) for the set whose elements are exactly \(a\), \(b\) and \(c\). We write \(\{x : P(x)\}\) or \(...