nLab Initiality Project - Semantics

Initiality Project - Semantics

Initiality Project - Semantics

This page is part of the Initiality Project.

Idea

See categorical model of dependent types, for now.

Basic definitions

Definition

A (natural-model-style) category with families (abbreviated CwF) consists of:

  • a (small) category CC
  • a terminal object 1C1\in C
  • two presheaves Tm,Ty[C op,Set]Tm,Ty \in [C^{op},Set]
  • a morphism of:TmTyof:Tm \to Ty
  • An algebraic representation for the map ofof, i.e. a function assigning to each morphism A:yΓTyA:y \Gamma \to Ty, where ΓC\Gamma\in C and yy denotes the Yoneda embedding, an object Γ.AC\Gamma.A\in C and a pullback square
    y(Γ.A) Tm yΓ A Ty.\array{ y(\Gamma.A) & \to & Tm \\ \downarrow &&\downarrow \\ y\Gamma & \xrightarrow{A} & Ty. }

Note that an algebraic representation for ofof can be equivalently expressed as choice of a right adjoint ext\mathsf{ext} for the corresponding functor of categories of elements el(Tm)el(Ty)el(Tm) \to el(Ty).

We consider categories with families as set-level structures (in particular, as strict categories), with morphism between them that preserve all the structure “on the nose” rather than merely up to isomorphism.

Definition

Let CC and DD be CwFs. A CwF morphism is a functor F:CDF:C\to D, preserving the specified terminal objects (on the nose), together with a commutative square in [C op,Set][C^{op},Set]:

Tm C F *(Tm D) Ty C F *(Ty D)\array{ Tm_C & \to & F^*(Tm_D) \\ \downarrow & & \downarrow \\ Ty_C & \to & F^*(Ty_D) }

such that, for all ΓC\Gamma \in C, and A:yΓTy CA : y\Gamma \to Ty_C, the square

y(F(Γ.A)) Tm D y(F(Γ)) Ty D \array{ y(F(\Gamma.A)) & \to & Tm_D \\ \downarrow & & \downarrow \\ y(F(\Gamma)) & \to & Ty_D }

is equal (on the nose) to the chosen pullback square corresponding to the morphism y(F(Γ))Ty Dy(F(\Gamma)) \to Ty_D.

The last condition can be reformulated in terms of categories of elements by requiring that the square of categories and functors

el(Ty C) el(Ty D) el(Tm C) el(Tm D) \array{ el(Ty_C) & \to & el(Ty_D) \\ \downarrow & & \downarrow \\ el(Tm_C) & \to & el(Tm_D) }

commutes strictly, where the vertical maps are the ext\mathsf{ext} functors corresponding to the chosen algebraic representations and the horizontal maps are el(F *)\el (F_*) the category of elements functor composed with the direct image functor applied to FF.

This defines a category CwF\mathbf{CwF} of categories with families and their morphisms, in which we aim to construct an initial object.

Type structure

Each type forming operation corresponds to structure on a CwF that can be expressed naturally in terms of categorical operations in the presheaf category [C op,Set][C^{op},Set].

Π\Pi-types

Given a presheaf TT on a CwF CC, let T Π(Γ)=Σ ATy(Γ)T(Γ.A)T^\Pi(\Gamma) = \Sigma_{A \in Ty(\Gamma)}T(\Gamma . A), so that Ty ΠTy^\Pi is the presheaf consisting of pairs of a type AA and a type BB in a context extended by AA and Tm ΠTm^\Pi is the presheaf of pairs of a type AA and a term in a context extended by AA.

Definition

Let CC be a CwF. A Π\Pi-type structure on CC consists of :

  • a natural transformation Π:Ty ΠTy\Pi : Ty^\Pi \to Ty
  • a natural transformation λ:Tm ΠTm\lambda : Tm^\Pi \to Tm
  • and a natural isomorphism
    ,:[C op,Set](,Ty Π)×[C op,Set](,Tm)[C op,Set](,Tm Π)\langle -, -\rangle : [C^{op}, Set]( -, Ty^\Pi) \times [C^{op}, Set](-, Tm) \to [C^{op}, Set](-, Tm^\Pi)

    exhibiting the following square as a pullback-square

    Tm Π λ Tm of Π of Ty Π Π Ty \array{ Tm^\Pi & \xrightarrow{\lambda} & Tm \\ of^\Pi \downarrow & & \downarrow of \\ Ty^\Pi & \xrightarrow{\Pi} & Ty }

Last revised on December 24, 2018 at 10:00:24. See the history of this page for a list of all contributions to it.