Contents

Idea

In type theory, a display map [Taylor 1983-7, §4.3.2] is a morphism $p\colon B\to A$ in a category which represents a dependent type under categorical semantics of dependent type theory.

That is, $B$ represents (interprets) a type dependent on a variable of type $A$, usually written syntactically as $x\colon A \vdash B(x)\colon Type$. The intended intuition is that $B(x)$ is the fiber of the map $p$ over $x\colon A$.

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

It follows automatically that the full subcategory on $D$ of the arrow category $\Arr(\mathcal{C})$ is replete; in other words, the property of $\mathcal{C}$-morphisms given by membership in $D$ respects the principle of equivalence.

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

This appears as (Taylor 1999, def. 8.3.2).

Examples

See also Taylor 1999, Exp. 8.3.6.

Related pages

• clan

• bundle

• categorical semantics of dependent type theory

• dominion

References

The notion is due to

• Paul Taylor, §4.3.2 in: Recursive Domains, Indexed Category Theory and Polymorphism, Cambridge (1983-7) [pdf]

• Paul Taylor, Section 8.3 in: Practical Foundations of Mathematics, Cambridge University Press (1999) [webpage]