natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
(see also Chern-Weil theory, parameterized homotopy theory)
In type theory, a display map is a morphism $p\colon B\to A$ in a category which represents a dependent type under some categorical semantics of type theory valued in that category.
That is, $B$ represents 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.
For $\mathcal{C}$ a category, a class $D \subset Mor(\mathcal{C})$ of morphisms of $\mathcal{C}$ is called a class of displays if
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 $D$ is not evil.
The category with displays is called well rooted if it has a terminal object $1$ and all the morphisms to $1$ are display maps.
It follows that binary products exist and their projections are display maps.
Often (which simplifies matters) $D$ is closed under composition:
Every identity morphism (and hence every isomorphism) belongs to $D$.
The composite of a composable pair of elements of $D$ belongs to $D$.
An interpretation or model of a category with class of displays $(\mathcal{C}, D)$ in another category with displays $(\mathcal{C}', D')$ is a functor
which preserves display maps and their pullbacks.
This appears as (Taylor, def. 8.3.2).
Examples of categories with displays, def. 1, include
One extreme: any locally cartesian category $\mathcal{C}$ with the class of all morphisms. (This is well rooted —if $\mathcal{C}$ has a terminal object— and closed under composition.)
Another extreme: any category with the empty class of no morphisms.
Modifying (2) for an extreme case closed under composition: any category with the class of isomorphisms.
Modifying (2) for an extreme well-rooted case: A category $\mathcal{C}$ with finite products and $D$ the class of product projections $X \times A \to X$. (This is also closed under composition.)
A locally cartesian category $\mathcal{C}$ (such as a topos) equipped with its class of monomorphisms. (This is closed under composition.)
A category equipped with a singleton pretopology. (This is closed under composition.)
(…)
See also Taylor, example 8.3.6.
Section 8.3 in