Spahn polynomial functor (Rev #2, changes)

which If we denote consider by$\mathrm{<span><del\; class="diffmod">\; P</del><ins\; class="diffmod">\; C</ins></span>}$ P C , is as called a semantics for a type theory we can ask howpolynomial functor$P=\Sigma_h \Pi_g f^*$ . If 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$g{\Sigma }_h=(-)\circ f$ g \Sigma_h=(-)\circ f , is this the preserves identity display we maps call if$\mathrm{<span><del\; class="diffmod">\; P</del><ins\; class="diffmod">\; h</ins></span>}$ P h is a display maps and these are closed under composition.linear polynomial functor$f^*$ . This forms construction pullbacks exists along e.g. if$\mathrm{<span><del\; class="diffmod">\; C</del><ins\; class="diffmod">\; f</ins></span>}$ C f , is and locally cartesian closed.$\Pi$ forms exponential objects. In particular for $P$ to preserve display maps it is sufficient that $g$ and $h$ are display maps.

If Initial algebras for polynomial functors, inductive families: A (weakly) initial algebra for an inductive polynomial functor we consider call an$C$inductive family . as An a inductive semantics family exists for a all type inductive theory polynomial we functors. 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 Examples: 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.

A (2a) (weakly) If initial algebra for an inductive polynomial functor we call aninductive family$X=\emptyset$ . An (the inductive corresponding family exists for all inductive polynomial functors. 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.

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