Spahn polynomial functor

Let $W\stackrel{f}{\leftarrow}X\stackrel{g}{\to}Y\stackrel{h}{\to}Z$ be a diagram in a category $C$; a diagram of this type is called a container or a indexed container. Let $f^*$, $\Pi_g$, $\Sigma$ denote base change resp. dependent product resp. dependent sum. Then the composite

$$C/W\stackrel{f^*}{\to}C/X\stackrel{\Pi_g}{\to}C/Y\stackrel{\Sigma_h}{\to}C/Z$$

which we denote by $P$, is called a polynomial functor. If $g$ is the identity we call $P$ a linear polynomial functor. This construction exists e.g. if $C$ is locally cartesian closed.

If we consider $C$ as a semantics for a type theory we can ask how $P=\Sigma_h \Pi_g f^*$ interacts with display maps: recall that by definition the class of display maps is closed under composition, pullbacks along arbitrary morphisms, and forming exponential objects. Since $\Sigma_h=(-)\circ f$, this preserves display maps if $h$ is a display maps and these are closed under composition. $f^*$ forms pullbacks along $f$, and $\Pi$ forms exponential objects. In particular for $P$ to preserve display maps it is sufficient that $g$ and $h$ are display maps.

A polynomial functor $\Sigma_h\Pi_g f^*$ where $g$ (but not necessarily $h$) is a display map, restricted to display maps, one could call a dependent polynomial functor or better an inductive polynomial functor.

Initial algebras for polynomial functors, inductive families: A (weakly) initial algebra for an inductive polynomial functor we call an inductive family. An inductive family exists for all inductive polynomial functors.

Examples:

(1) If $h=\Delta:Z\to Z\times Z$ is a display map, then $Id_Z(z,z)=\{refl\}$

(2a) If $X=\emptyset$ (the corresponding polynomial functor $P$ is then constant) and $h$ is a display map, then the initial algebra of the polynomial functor is $h$. In particular the functor for identity types is of this form.

(2b) If in (2a) $h$ is not a display map, then the initial algebra of $P$ is a map $\hat h:\hat A\to Z$ equipped with a map $\check{h}:h\to \hat h$ over $Z$ with the property that every other map $h\to q$ over $Z$ with $q$ a display map, factors through $\hat h$.

The Coq defining an inductive family is

Inductive hfiber {A X} (I: A ->X) : (X->Type):=
inj : forall (x:A), hfiber I(Ix).

It is just the homotopy fiber of $I$. In particular the identity type of $X$ is the homotopy fiber of the diagonal $\Delta:X\to X\times X$.