hcju/proofwikips huggingface dataset:
Definition:Restricted Existential Quantifier
Let $A$ be a class in ZF. The '''restricted existential quantifier''' is denoted $\exists x \in A$ and is defined as the following definitional abbreviation: :$\exists x \in A: P \left({x}\right) \quad \text{for} \quad \exists x: \left({x \in A \land P \left({x}\right)}\right)$ where $P \left({x}\right)$ is any well-formed formula of the language of set theory.
Definition:Language of Set Theory
 The language of set theory consists of the language of predicate logic with the binary predicate symbol $\in$, denoting membership. === Predicate Symbols === The language of set theory uses only one predicate symbol, $\in$, the membership sign. It is a binary predicate symbol. Using $\in$, other symbols such as $=$ can be defined (see Definition:Set Equality). === Connectives === The language of set theory borrows the connectives from the language of predicate logic.  However, some of the connectives can be considered definitional abbreviations. $\land$ and $\neg$ can be taken as primitive connectives with the following definitional abbreviations: {{begin-eqn}} {{eqn | l = P \implies Q       | o = \operatorname{for}       | r = \neg ( P \land \neg Q )       | c =  }} {{eqn | l = P \lor Q       | o = \operatorname{for}       | r = \neg ( \neg P \land \neg Q )       | c =  }} {{eqn | l = P \iff Q       | o = \operatorname{for}       | r = ( ( P \implies Q ) \land ( Q \implies P ) )       | c = justified because $\implies$ is already a definitional abbreviation }} {{end-eqn}} $\implies$ and $\neg$ can be taken as primitive connectives with the following definitional abbreviations: {{begin-eqn}} {{eqn | l = ( P \land Q )       | o = \operatorname{for}       | r = \neg ( P \implies \neg Q )       | c =  }} {{eqn | l = ( P \lor Q )       | o = \operatorname{for}       | r = ( \neg P \implies Q )       | c =  }} {{eqn | l = ( P \iff Q )       | o = \operatorname{for}       | r = \neg ( ( P \implies Q ) \implies \neg ( Q \implies P ) )       | c =  }} {{end-eqn}} The Sheffer stroke $\mid$ can be taken as a sole primitive connective with the following definitional abbreviations: {{begin-eqn}} {{eqn | l = \neg P       | o = \operatorname{for}       | r = ( P \mid P )       | c =  }} {{eqn | l = ( P \implies Q )       | o = \operatorname{for}       | r = ( P \mid ( Q \mid Q ) )       | c =  }} {{end-eqn}} The other connectives can be defined using $\neg P$ and $( P \implies Q )$ as the "new" primitive connectives. === Quantifiers === The language of set theory adopts the same quantifiers as those in the language of predicate logic. However, only $\forall$ is necessary to adopt as a primitive symbol, and $\exists$ can be defined: :$\exists x: P(x) \operatorname{for} \neg \forall x: \neg P(x)$
Definition:Russell Class
The '''Russell class''', denoted $\operatorname{Ru}$, equals the class of all sets $x$ that are not elements of themselves: :$\operatorname{Ru} = \set {x: x \notin x}$
Definition:Class Equality
Let $A$ and $B$ be classes. === Definition 1 === {{:Definition:Class Equality/Definition 1}} === Definition 2 === {{:Definition:Class Equality/Definition 2}}
Definition:Class/Zermelo-Fraenkel
A '''class''' in $\textrm{ZF}$ is a formal vehicle capturing the intuitive notion of a class, namely a collection of all sets such that a particular condition $P$ holds. In $\textrm{ZF}$, '''classes''' are written using class builder notation: :$\set {x : \map P x}$ {{Questionable|Wouldn't you need the Class Comprehension Axiom for this?}} where $\map P x$ is a statement containing $x$ as a free variable.   More formally, a '''class''' $\set {x: \map P x}$ serves to define the following definitional abbreviations involving the membership symbol: {{begin-eqn}} {{eqn | l = y \in \set {x: \map P x}       | o = \quad \text{for} \quad       | r = \map P y }} {{eqn | l = \set {x: \map P x} \in y       | o = \quad \text{for} \quad       | r = \exists z \in y: \forall x: \paren {x \in z \iff \map P x} }} {{eqn | l = \set {x: \map P x} \in \set {y: \map Q y}       | o = \quad \text{for} \quad       | r = \exists z: \paren {\map Q z \land \forall x: \paren {x \in z \iff \map P x} } }} {{end-eqn}} where: :$x, y ,z$ are variables of $\textrm{ZF}$ :$P, Q$ are propositional functions. Through these "rules", every statement involving $\set {x: \map P x}$ can be reduced to a simpler statement involving only the basic language of set theory.
Definition:Small Class
Let $A$ denote an arbitrary class. Then $A$ is said to be '''small''' : :$\exists x: x = A$ where $=$ denotes class equality and $x$ is a set variable. That is, a class is '''small'''  it is equal to some set variable. To denote that a class $A$ is '''small''', the notation $\mathcal M \left({A}\right)$ may be used. Thus: : $\mathcal M \left({A}\right) \iff \exists x: x = A$
Definition:Set Variable
In the language of set theory, '''set variables''' are the variables. '''Set variables''' are sometimes referred to as sets.
Definition:Unordered Tuple
An '''unordered tuple''' is a small class in Zermelo-Fraenkel set theory that contains a finite number of elements. It is written using roster notation and is denoted: :$\set {a_1, a_2, \ldots, a_n}$ that is: :$\set {a_1, a_2, \ldots, a_n} = \set {x: x = a_1 \lor x = a_2 \lor \dots \lor x = a_n}$
Definition:Category of Sets
The '''category of sets''', denoted $\mathbf{Set}$ or $\mathbf{Sets}$ is the metacategory with: {{DefineCategory |ob   = sets |mor  = mappings |comp = composition of mappings |id   = identity mappings }}
Definition:Category of Finite Sets
The '''category of finite sets''', denoted $\mathbf{Finset}$ or $\mathbf{Sets}_{\text{fin}}$ is the metacategory with: {{DefineCategory |ob  = All finite sets; |mor = All mappings between finite sets. }}
Definition:Category of Ordered Sets
The '''category of ordered sets''', denoted $\mathbf{OrdSet}$, is the metacategory with: {{DefineCategory | ob = ordered sets | mor = increasing mappings | comp = composition of mappings | id = identity mappings }}
Definition:Category of Relations
The '''category of (binary) relations''', denoted $\mathbf{Rel}$, is the metacategory with: {{DefineCategory |ob   = sets |mor  = binary relations $\mathcal R \subseteq A \times B$. |comp = composition of relations |id   = identity mappings }}
Definition:One (Category)
The '''category one''' $\mathbf 1$, is the category with: {{DefineCategory |ob = One object: $*$ |mor = One morphism: the identity morphism $\operatorname{id}_*$. }}
Definition:Two (Category)
The category $\mathbf 2$, '''two''', is the category: :$\quad * \longrightarrow \star$ with: :Two objects, $*$ and $\star$  :Three morphisms: ::The identity morphism $I_*$ ::The identity morphism $I_\star$ ::One non-identity morphism $* \to \star$
Definition:Three (Category)
The category $\mathbf 3$, '''three''', is the category: ::$\begin{xy} <0em,0em>*+{\circledast} = "x", <5em,0em>*+{\star} = "y", <5em,-5em>*+{\bullet} = "z", "x";"y" **@{-} ?>*@{>}, "y";"z" **@{-} ?>*@{>}, "x";"z" **@{-} ?>*@{>} \end{xy}$ with: :Three objects, $\circledast$, $\star$ and $\bullet$; and  :Six morphisms: :* The three identity morphisms $\operatorname{id}_\circledast$, $\operatorname{id}_\star$, $\operatorname{id}_\bullet$; :* One non-identity morphism $\circledast \to \star$; :* One non-identity morphism $\star \to \bullet$; :* One non-identity morphism $\circledast \to \bullet$.
Definition:Zero (Category)
The category $\mathbf 0$, '''zero''', is the empty category: :$\qquad$ with: :No objects and consequently: :No morphisms.
Definition:Functor/Covariant
Let $\mathbf C$ and $\mathbf D$ be metacategories. A '''covariant functor''' $F: \mathbf C \to \mathbf D$ consists of: * An '''object functor''' $F_0$ that assigns to each object $X$ of $\mathbf C$ an object $FX$ of $\mathbf D$. * An '''arrow functor''' $F_1$ that assigns to each arrow $f: X \to Y$ of $\mathbf C$ an arrow $Ff : FX \to FY$ of $\mathbf D$. These '''functors''' must satisfy, for any morphisms $X \stackrel f \longrightarrow Y \stackrel g \longrightarrow Z$ in $\mathbf C$: :$\map F {g \circ f} = F g \circ F f$ and: :$\map F {\operatorname {id}_X} = \operatorname{id}_{F X}$ where $\operatorname {id}_W$ denotes the identity arrow on an object $W$, and $\circ$ is the composition of morphisms. The behaviour of a '''covariant functor''' can be pictured as follows: ::$\begin{xy} <4em,4em>*{\mathbf C} = "C", <0em,0em>*+{X} = "a", <4em,0em>*+{Y} = "b", <4em,-4em>*+{Z}= "c", "a";"b" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f}, "b";"c" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g}, "a";"c" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{g \circ f}, "C"+/r9em/*{\mathbf D}, "C"+/r2em/;"C"+/r6em/ **@{-} ?>*@{>} ?*!/_1em/{F}, "b"+/r2em/+/_2em/;"b"+/r6em/+/_2em/ **@{~} ?>*@2{>} ?<>(.5)*!/_.6em/{F}, "a"+/r13em/*+{FX}="Fa",  "b"+/r13em/*+{FY}="Fb",  "c"+/r13em/*+{FZ}="Fc", "Fa";"Fb" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{Ff}, "Fb";"Fc" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{Fg}, "Fa";"Fc" **@{-} ?>*@{>} ?<>(.7)*!/r3em/{F \left({g \circ f}\right) = \\ Fg \circ Ff}, \end{xy}$
Definition:Functor/Contravariant
{{:Definition:Functor/Contravariant/Definition 1}}
Definition:Composition of Functors
Let $\mathbf C, \mathbf D$ and $\mathbf E$ be metacategories. Let $F: \mathbf C \to \mathbf D$ and $G: \mathbf D \to \mathbf E$ be (covariant) functors. The '''composition of $G$ with $F$''' is the functor $GF: \mathbf C \to \mathbf E$ defined by: :For all objects $C$ of $\mathbf C$: $\hskip{2.9cm} GF \left({C}\right) := G \left({FC}\right)$ :For all morphisms $f: C_1 \to C_2$ of $\mathbf C$: $\quad GF \left({f}\right) := G \left({Ff}\right)$ $GF$ is said to be a '''composite functor'''.
Definition:Identity Functor
Let $\mathbf C$ be a metacategory. The '''identity functor on $\mathbf C$''' is the functor $\operatorname{id}_{\mathbf C}: \mathbf C \to \mathbf C$ defined by: :For all objects $C$ of $\mathbf C$: $\hskip{2.9cm} \operatorname{id}_{\mathbf C} C := C$ :For all morphisms $f: C_1 \to C_2$ of $\mathbf C$: $\quad \operatorname{id}_{\mathbf C} f := f$ That $\operatorname{id}_{\mathbf C}$ constitutes a functor is shown on Identity Functor is Functor.
Definition:Category of Categories
The '''category of categories''', denoted $\mathbf{Cat}$, is the metacategory with: {{DefineCategory |ob = small categories |mor = functors |comp = composition of functors |id = identity functors }}
Definition:Preorder Category
=== Definition 1 === {{:Definition:Preorder Category/Definition 1}} === Definition 2 === {{:Definition:Preorder Category/Definition 2}}
Definition:Order Category
=== Definition 1 === {{:Definition:Order Category/Definition 1}} === Definition 2 === {{:Definition:Order Category/Definition 2}}
Definition:Discrete Category
Let $\CC$ be a metacategory. Then $\CC$ is said to be '''discrete'''  it comprises only identity morphisms. If the collection $\CC$ constitutes the objects of $\mathbf C$, then $\mathbf C$ may also be denoted $\map {\mathbf {Dis} } \CC$.
Definition:Monoid Category
Let $\left({S, \circ}\right)$ be a monoid with identity $e_S$. One can interpret $\left({S, \circ}\right)$ as being a category, with: {{DefineCategory | ob = Only one, say $*$ | mor = $a: * \to *$, for all $a \in S$ | comp =  $a \circ b: * \to *$ is defined using the operation $\circ$ of the monoid $S$ | id = $\operatorname{id}_* := e_S: * \to *$ }} The category that so arises is called a '''monoid category'''.
Definition:Category of Monoids
The '''category of monoids''', denoted $\mathbf{Mon}$, is the metacategory with: {{DefineCategory | ob = monoids | mor = monoid homomorphisms }}
Definition:Monoid Homomorphism
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be monoids. Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$. That is, $\forall a, b \in S$: :$\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$ Suppose further that $\phi$ preserves identities, i.e.: :$\phi \left({e_S}\right) = e_T$ Then $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ is a monoid homomorphism.
Definition:Prime Power
A '''prime power''' $p^n$ is a positive integer which is a prime number $p$ raised to the power $n$. The list of prime powers begins: :$1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, \ldots$ {{OEIS|A000961}}
Definition:Locally Small Category
Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to be '''locally small''' iff all of its hom classes are sets. That is, iff for all objects $X, Y \in \mathbf C_0$ of $\mathbf C$: :$\operatorname{Hom}_{\mathbf C} \left({X, Y}\right) = \left\{{f \in \mathbf C_1 }\ \middle\vert \ {f: X \to Y}\right\}$ is a set.
Definition:Projection Functor
Let $\mathbf C$ and $\mathbf D$ be metacategories, and let $\mathbf C \times \mathbf D$ be their product. The '''first projection functor''' $\operatorname{pr}_1: \mathbf C \times \mathbf D \to \mathbf C$ is defined by: :$\operatorname{pr}_1 \left({C, D}\right) := C$ for all objects $\left({C, D}\right) \in \operatorname{ob} \mathbf C \times \mathbf D$ :$\operatorname{pr}_1 \left({f, g}\right) := f$ for all morphisms $\left({f, g}\right) \in \operatorname{mor} \mathbf C \times \mathbf D$ The '''second projection functor''' $\operatorname{pr}_2: \mathbf C \times \mathbf D \to \mathbf D$ is defined by: :$\operatorname{pr}_2 \left({C, D}\right) := D$ for all objects $\left({C, D}\right) \in \operatorname{ob} \mathbf C \times \mathbf D$ :$\operatorname{pr}_2 \left({f, g}\right) := g$ for all morphisms $\left({f, g}\right) \in \operatorname{mor} \mathbf C \times \mathbf D$ That these constitute functors is shown on Projection Functor is Functor.
Definition:Dual Category
Let $\mathbf C$ be a metacategory. Its '''dual category''', denoted $\mathbf C^{\text{op} }$, is defined as follows: {{DefineCategory |ob = $X^{\text{op} }$, for all $X \in \operatorname{ob}\mathbf C$ |mor = $f^{\text{op} }: D^{\text{op} } \to C^{\text{op} }$ for all $f: C \to D$ in $\mathbf C_1$ |comp = $\left({f^{\text{op} } \circ g^{\text{op} } }\right) := \left({g \circ f}\right)^{\text{op} }$, whenever this is defined |id = $\operatorname{id}_{X^{\text{op} } } := \operatorname{id}_X^{\text{op} }$ }}
Definition:Morphism Category
Let $\mathbf C$ be a metacategory. Its '''morphism category''', denoted $\mathbf C^\to$, is defined as follows: {{DefineCategory |ob = The morphisms $\mathbf C_1$ of $\mathbf C$ |mor = $g: f \to f'$ is a pair $\left({g_1, g_2}\right)$ of morphisms of $\mathbf C$ such that $g_2 \circ f = f' \circ g_1$ |comp = $\left({h_1, h_2}\right) \circ \left({g_1, g_2}\right) := \left({h_1 \circ g_1, h_2 \circ g_2}\right)$, whenever this is defined |id = $\operatorname{id}_{f} := \left({\operatorname{id}_C, \operatorname{id}_D}\right)$ for $f: C \to D$ }}
Definition:Domain Functor
Let $\mathbf C$ be a metacategory. Let $\mathbf C^\to$ be its morphism category. The '''domain functor''' is the functor $\operatorname{\mathbf{dom}}: \mathbf C^\to \to \mathbf C$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \operatorname{\mathbf{dom} } f := \operatorname{dom} f }} {{axiom|lc= Morphism functor:        |m = \operatorname{\mathbf{dom} } \left({g_1, g_2}\right) := g_1 }} {{end-axiom}} That it is in fact a functor is shown on Domain Functor is Functor. The functor $\mathbf{dom}$ can be represented as follows: ::$\begin{xy} <0em,2em>*+{A} = "A", <0em,-2em>*+{B} = "B", <4em,2em>*+{A'} = "A2", <4em,-2em>*+{B'} = "B2", "A";"B" **@{-} ?>*@{>} ?*!/^1em/{f}, "A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, "A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'}, "B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2}, <6em,0em>;<10em,0em> **@{~} ?>*@2{>} ?*!/_1em/{\mathbf{dom}}, <12em,2em>*+{A} = "AA", <16em,2em>*+{A'} = "AA2", "AA";"AA2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, \end{xy}$ It is thus seen to be an example of a forgetful functor.
Definition:Codomain Functor
Let $\mathbf C$ be a metacategory. Let $\mathbf C^\to$ be its morphism category. The '''codomain functor''' is the functor $\operatorname{\mathbf{cod}}: \mathbf C^\to \to \mathbf C$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \operatorname{\mathbf{cod} } f := \operatorname{cod} f }} {{axiom|lc= Morphism functor:        |m = \operatorname{\mathbf{cod} } \left({g_1, g_2}\right) := g_2 }} {{end-axiom}} That it is in fact a functor is shown on Codomain Functor is Functor. The functor $\mathbf{cod}$ can be represented as follows: ::$\begin{xy} <0em,2em>*+{A} = "A", <0em,-2em>*+{B} = "B", <4em,2em>*+{A'} = "A2", <4em,-2em>*+{B'} = "B2", "A";"B" **@{-} ?>*@{>} ?*!/^1em/{f}, "A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, "A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'}, "B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2}, <6em,0em>;<10em,0em> **@{~} ?>*@2{>} ?*!/_1em/{\mathbf{cod}}, <12em,-2em>*+{B} = "BB", <16em,-2em>*+{B'} = "BB2", "BB";"BB2" **@{-} ?>*@{>} ?*!/^1em/{g_2}, \end{xy}$ It is thus seen to be an example of a forgetful functor.
Definition:Slice Category
Let $\mathbf C$ be a metacategory. Let $C \in \mathbf C_0$ be an object of $\mathbf C$. The '''slice category''' of $\mathbf C$ over $C$, denoted $\mathbf C / C$, is defined as follows: {{DefineCategory |ob = $f: X \to C$, i.e. the morphisms of $\mathbf C$ with codomain $C$ |mor = $a: f \to f'$, for all morphisms $a \in \mathbf C_1$ with $f' \circ a = f$ |comp = $a \circ b$ is defined precisely as in $\mathbf C$ |id = $\operatorname{id}_f := \operatorname{id}_X$, for $f: X \to C$ }} The morphisms can be displayed using a commutative diagram as follows: ::$\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,-4em>*+{C} = "C", "X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a}, "X";"C" **@{-} ?>*@{>} ?<>(.3)*!/^1em/{f}, "X2";"C" **@{-} ?>*@{>} ?<>(.3)*!/_1em/{f'}, \end{xy}$
Definition:Permutation on n Letters/Cycle Notation
The $k$-cycle $\rho$ is denoted: :$\begin {pmatrix} i & \map \rho i & \ldots & \map {\rho^{k - 1} } i \end{pmatrix}$ From Existence and Uniqueness of Cycle Decomposition, all permutations can be defined as the product of disjoint cycles. As Disjoint Permutations Commute, the order in which they are performed does not matter. So, for a given permutation $\rho$, the '''cycle notation''' for $\rho$ consists of all the disjoint cycles into which $\rho$ can be decomposed, concatenated as a product. It is conventional to omit $1$-cycles from the expression, and to write those cycles with lowest starting number first.
Definition:Signum Function/Natural Numbers
The '''signum function''' $\sgn: \N \to \set {0, 1}$ is the restriction of the signum function to the natural numbers, defined as: :$\forall n \in \N: \map \sgn n := \begin{cases} 0 & : n = 0 \\ 1 & : n > 0 \end{cases}$
Definition:Signum Function/Signum Complement
Let $\sgn: \N \to \set {0, 1}$ be the signum function on the natural numbers. The '''signum complement function''' $\overline \sgn: \N \to \set {0, 1}$ is defined as: :$\forall n \in \N: \map {\overline \sgn} n := \begin{cases} 1 & : n = 0 \\ 0 & : n > 0 \end{cases}$
Definition:Sign of Permutation
Let $n \in \N$ be a natural number. Let $\N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$. Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers. Let $\pi$ be a permutation of $\N_n$. Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$. Let $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be defined as: :$\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n} := \map {\Delta_n} {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$ The '''sign of $\pi \in S_n$''' is defined as: :$\map \sgn \pi = \begin{cases} \dfrac {\Delta_n} {\pi \cdot \Delta_n} & : \Delta_n \ne 0 \\ 0 & : \Delta_n = 0 \end{cases}$
Definition:Composition Functor on Slice Categories
Let $\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\mathbf C$. Let $\mathbf C / C$ and $\mathbf C / D$ be the associated slice categories. Let $g: C \to D$ be a morphism of $\mathbf C$. Then $g$ defines a '''composition functor''' $g_* : \mathbf C / C \to \mathbf C / D$: {{begin-axiom}} {{axiom|lc= Object functor:        |m = g_* f := g \circ f        |rc= The composition $\circ$ is taken in $\mathbf C$ }} {{axiom|lc= Morphism functor:        |m = g_* a := a }} {{end-axiom}} That it is in fact a functor is shown on Composition Functor on Slice Categories is Functor. The effect of $g_*$ is captured in the following commutative diagram: ::$\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,-4em>*+{C} = "C", <0em,-8em>*+{D} = "D", "X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a}, "X";"C" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{f}, "X2";"C" **@{-} ?>*@{>} ?<>(.4)*!/_.6em/{f'}, "C";"D" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{g}, "X";"D" **\crv{<-5em,-4em>} ?>*@{>} ?*!/^1.6em/{g_* f = \\ g \circ f}, "X2";"D" **\crv{<5em,-4em>} ?>*@{>} ?*!/_1.6em/{g_* f' = \\ g \circ f'}, \end{xy}$
Definition:Slice Functor
Let $\mathbf C$ be a metacategory. Let $\mathbf{Cat}$ be the category of categories. The '''slice functor''' is the functor $\mathbf C / \cdot: \mathbf C \to \mathbf{Cat}$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \mathbf C / C := \mathbf C / C }} {{axiom|lc= Morphism functor:        |m = \mathbf C / f := f_* }} {{end-axiom}} where $\mathbf C / C$ is a slice category, and $f_*$ is the composition functor defined by $f$. The effect of $\mathbf C / \cdot$ is captured in the following diagram: ::$\begin{xy} <0em,0em>*+{A} = "a", <4em,0em>*+{B} = "b", <4em,-4em>*+{C}= "c", "a";"b" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f}, "b";"c" **@{-} ?>*@{>} ?<>(.5)*!/_.6em/{g}, "a";"c" **@{-} ?>*@{>} ?<>(.4)*!/^1em/{g \circ f}, "b"+/r4em/+/_3em/;"b"+/r8em/+/_3em/ **@{~} ?>*@2{>} ?*!/_1em/{\mathbf C / \cdot}, "a"+/r13em/*+{\mathbf C / A}="Fa",  "b"+/r14em/*+{\mathbf C / B}="Fb",  "c"+/r14em/+/_1em/*+{\mathbf C / C}="Fc", "Fa";"Fb" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f_*}, "Fb";"Fc" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g_*}, "Fa";"Fc" **@{-} ?>*@{>} ?<>(.7)*!/r3em/{\left({g \circ f}\right)_* \\ = g_* f_*}, \end{xy}$ where $g_* f_*$ denotes a composite functor.
Definition:Coslice Category
Let $\mathbf C$ be a metacategory. Let $C \in \mathbf C_0$ be an object of $\mathbf C$. The '''coslice category''' of $\mathbf C$ under $C$, denoted $C / \mathbf C$, is the category with: {{DefineCategory |ob = $f: C \to X$, i.e. the morphisms of $\mathbf C$ with domain $C$ |mor  = $a: f \to f'$, for all morphisms $a \in \mathbf C_1$ with $a \circ f' = f$ |comp = $a \circ b$ is defined precisely as in $\mathbf C$ |id = $\operatorname{id}_f := \operatorname{id}_X$, for $f: C \to X$ }} The morphisms can be displayed using a commutative diagram as follows: ::$\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,4em>*+{C} = "C", "X";"X2" **@{-} ?>*@{>} ?*!/^1em/{a}, "C";"X" **@{-} ?>*@{>} ?*!/^.6em/{f}, "C";"X2" **@{-} ?>*@{>} ?<>(.6)*!/_1em/{f'}, \end{xy}$
Definition:Category of Pointed Sets
The '''category of pointed sets''', denoted $\mathbf {Set}_*$, is defined as follows: {{DefineCategory | ob = pointed sets $\struct {A, a}$ | mor = pointed mappings | comp = Standard composition of mappings | id = $\map {\mathrm {id}_{\struct {A, a} } } := \mathrm {id}_A$, the identity mapping on $A$ }}
Definition:Kleene Closure
Let $S$ be a set. The '''Kleene closure''' of $S$, denoted $S^*$, is the set of all ordered tuples in $S$.
Definition:Concatenation of Ordered Tuples
Let $S$ be a set. Let $w, w'$ be finite sequences in $S$ of lengths $n$ and $n'$, respectively. Then the '''concatenation of $w$ and $w'$''', denoted $w * w'$ or simply $w w'$, is the sequence of $n + n'$ terms defined by: :$\map {w * w'} i := \begin{cases} \map w i & : \text {if $1 \le i \le n$} \\ \map {w'} {i - n} & : \text {if $n < i \le n + n'$} \end{cases}$ === Algebraic Structure ===  Let $S^*$ be the Kleene closure of $S$. Then $\struct {S^*, *}$ is an algebraic structure, and by definition a magma.
Definition:Insertion of Generators
Let $S$ be a set, and let $S^*$ be its Kleene closure. The '''insertion of generators (into $S^*$)''' is the mapping $i: S \to S^*$ defined by: :$i \left({s}\right) := \left\langle{s}\right\rangle$ that is, it sends any element $s$ of $S$ to the one-term sequence containing only $s$.