nLab notation

Idea

Notation is the assigning of abstract stuff, structure, and properties a typographic “symbol”. The range of the terms, definitions, and glossaries is intended to be indepth, including various synonyms to ensure coverage.

Glossary

• $\wedge$ The null-ordered logic conjunction, or AND operator.

• $\wedge _A$ The meet operator in the meet-semilattice A.

• $\vee$ The null-ordered logic inclusive dysjunction, or OR operator.

• $\vee _A$ The join operator in the join-semilattice A.

• $\veebar$ The null-ordered logic exclusive dysjunction, or XOR operator.

• $\Rightarrow$ The null-ordered logic implication, conditional, or implies operator. For propositions $\phi, \psi$,

Theorem

$\phi \Rightarrow \psi = \neg \phi \vee \psi$

• $\forall$ The first-order logic universal quantifier. For predicate formula $\phi$, $\forall \alpha \phi (\alpha) = \neg \exists \alpha \neg \phi (\alpha)$.

• $\exists$ The first-order logic existential quantifier.

• $\exists !$ The first-order logic uniqueness quantifier.

• $\uparrow _A$ $(${$\alpha$}$)$ The principal up-set, coinitial/final segment generated by the singleton subset {$\alpha$} in the partial order, or join-semilattice A.

• $\uparrow _A (\alpha)$ The up-set, or coinitial/final segment generated by the subset $\alpha$ in the partial order, or join-semilattice A.

• $0$ The first limit-ordinal in the Von Neumann ordinals, i.e. ∅. The initial category.

• $0_A$ The bottom element, or the zero of complete partial order, or bound meet-semilattice $A$.

• $1$ The first successor-ordinal in the Von Neumann ordinals, i.e. ∅. The terminal category.

• $1_A$ The top element, or unit of the complete partial order, or bound meet-semilattice $A$.

• $Fun_A(B,C), Hom_A(B,C)$ The hom-set, or hom-object called the functor category of functors from domain B, to codomain C in category A.

• $Hom_A(x,y), hom(x,y), A(x,y)$ The hom-set, or hom-object of morphisms from domain x, and codomain y in category A.

• $Nat _A(f,g)$, $Hom_A(f,g)$ The hom-set or hom-object of natural transformations from domain f, and codomain g in functor category A.

• ZFC The standard first-order foundational material pure axiomatic set theoretical logic with Axiom of Choice.

Not at all complete.*