nLab
display map

Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination? for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Bundles

Contents

Idea

In type theory, a display map is a morphism p:BAp\colon B\to A in a category which represents a dependent type under some categorical semantics of type theory valued in that category.

That is, BB represents a type dependent on a variable of type AA, usually written syntactically as x:AB(x):Typex\colon A \vdash B(x)\colon Type. The intended intuition is that B(x)B(x) is the fiber of the map pp over x:Ax\colon A.

More specifically, the syntactic category of a type theory is naturally equipped with a class of maps called “display maps” which come from its type dependencies, while models for such a theory can be studied in any other category equipped with a suitable class of maps called “display maps.” By the usual adjunction, such models are then equivalent to functors out of the syntactic category which preserve the relevant structure, including the display maps.

Since the most common dependent type theories include dependent product types, and any display map in a category that represents semantics for such a theory must be exponentiable, the term display map is sometimes used to mean any exponentiable morphism.

Sometimes all maps are display maps. This happens particularly in extensional type theory, such as versions of the internal logic of categories that include dependent types.

In the context of homotopy type theory display maps are fibrations, for instance in a type-theoretic model category.

Definition

Definition

For 𝒞\mathcal{C} a category, a class DMor(𝒞)D \subset Mor(\mathcal{C}) of morphisms of 𝒞\mathcal{C} is called a class of displays if

  • All pullbacks of elements of DD exist and belong to DD.

It follows automatically that the full subcategory on DD of the arrow category Arr(𝒞)\Arr(\mathcal{C}) is replete; in other words, the property of 𝒞\mathcal{C}-morphisms given by membership in DD is not evil.

Definition

The category with displays is called well rooted if it has a terminal object 11 and all the morphisms to 11 are display maps.

It follows that binary products exist and their projections are display maps.

Definition

Often (which simplifies matters) DD is closed under composition:

  1. Every identity morphism (and hence every isomorphism) belongs to DD.

  2. The composite of a composable pair of elements of DD belongs to DD.

Definition

An interpretation or model of a category with class of displays (𝒞,D)(\mathcal{C}, D) in another category with displays (𝒞,D)(\mathcal{C}', D') is a functor

S:𝒞𝒞 S\colon \mathcal{C} \to \mathcal{C}'

which preserves display maps and their pullbacks.

This appears as (Taylor, def. 8.3.2).

Examples

Examples

Examples of categories with displays, def. 1, include

  1. One extreme: any locally cartesian category 𝒞\mathcal{C} with the class of all morphisms. (This is well rooted —if 𝒞\mathcal{C} has a terminal object— and closed under composition.)

  2. Another extreme: any category with the empty class of no morphisms.

  3. Modifying (2) for an extreme case closed under composition: any category with the class of isomorphisms.

  4. Modifying (2) for an extreme well-rooted case: A category 𝒞\mathcal{C} with finite products and DD the class of product projections X×AXX \times A \to X. (This is also closed under composition.)

  5. A locally cartesian category 𝒞\mathcal{C} (such as a topos) equipped with its class of monomorphisms. (This is closed under composition.)

  6. A category equipped with a singleton pretopology. (This is closed under composition.)

  7. (…)

See also Taylor, example 8.3.6.

References

Section 8.3 in

Revised on September 15, 2013 18:25:30 by Urs Schreiber (89.204.137.43)