| ([[nLab:(infinity,1)-category theory|homotopy]]) [[nLab:category theory]] | ([[nLab:homotopy type theory|homotopy]]) [[nLab:type theory]] | coq | |----------------------|------------------|------| | [[nLab:semantics]] | [[nLab:syntax]] | identifier | | [[nLab:object classifier]] $\Omega$| Type | `Type` | | [[nLab:object]] $X$ | [[nLab:type]] $x : X$ | | | [[nLab:fibration]]([[nLab:display map]]) $\array{A \\ \downarrow^{\mathrlap{p}} \\ X}$ | [[nLab:dependent type]] $x : X \vdash A(x) : Type$ | `x : X - P x : Type`| | [[nLab:section]] $\array{ X &&\stackrel{t}{\to}&& A \\ & {}_{\mathllap{id}}\searrow && \swarrow_{p} \\ && X}$ | [[nLab:term]] $x : X \vdash t(x) : A(x)$ | | | [[nLab:pullback]] $\array{ f^* A &\to& A \\ \downarrow && \downarrow^{\mathrlap{p}} \\ Y &\stackrel{f}{\to} & X }$ | [[nLab:substitution]] $y : Y \vdash A(f(y)) : Type$ | | | [[nLab:direct image]] $\array{ A && f_* A \\ {}^{\mathllap{p}}\downarrow && \downarrow ^{f_* p}\\ X &\stackrel{f}{\to}& Y}$ | [[nLab:dependent product]] $ y : Y \vdash \underset{x : X(y)}{\prod } A(x) : Type$ | | | [internal hom in slice](locally%20cartesian%20closed%20category#EquivalentCharacterizations) $\array{ X \times f^* A && f_* f^* A & = [X,A]_Y \\ {}^{\mathllap{}}\downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y}$ | [[nLab:function type]] $ y : Y \vdash X(y) \to A(y) : Type$ | | | postcomposition $\array{ A &=& f_! A \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y}$ | [[nLab:dependent sum]] $y : Y \vdash \underset{x : X(y)}{\sum} A(x) : Type$ | `{ x : X & P x } : Type` or `sig T (fun (x : X) => P x) : Type.` or `sig T P : Type` | | [[nLab:fiber product|fiber]][[nLab:product]] $\array{ X \times f^* A &=& f_! f^* A & = X \times_Y A\\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y}$ |[[nLab:product type]] $y : Y \vdash X(y) \times A(y) : Type$ | | | [[nLab:Beck-Chevalley condition]] of [[nLab:codomain fibration]] | [[nLab:substitution]] commutes with [[nLab:dependent sum]] | | | [[nLab:path space object]] $\array{A^I \\ \downarrow^{\mathrlap{(d_1,d_0)}} \\ A \times A}$ | [[nLab:identity type]] $a,b : A \vdash (a = b)$ | | | [[nLab:truncated object in an (infinity,1)-category|(-2)-truncated morphism]]/[[nLab:equivalence in an (infinity,1)-category|equivalence]] $\array{Y \\ \downarrow^{\mathrlap{\simeq}} \\ X}$ | [[nLab:true]]/[[nLab:unit type]] $x : X \vdash 1 : Type$ | | | [[nLab:truncated object in an (infinity,1)-category|(-1)-truncated morphism]]/[[nLab:monomorphism in an (infinity,1)-category|monomorphism]] $\array{\phi \\ \downarrow \\ X}$ | [[nLab:proposition]] $x : X \vdash \phi(x) : Type$ | | | [[nLab:direct image]] of [[nLab:truncated object in an (infinity,1)-category|(-1)-truncated morphism]] | [[nLab:universal quantifier]] $y : Y \vdash \underset{x \in X(y)}{\forall} \phi(x) : Type$ | `forall x : X, P x`| | [[nLab:truncated object of an (infinity,1)-category|(-1)-truncation]] of postcomposition of [[nLab:truncated object in an (infinity,1)-category|(-1)-truncated morphism]] | [[nLab:existential quantifier]] $y : Y \vdash \underset{x \in X(y)}{\exists} \phi(x) $ | `exists x : X, P x` | | | | |---|---| | text | text |