Title: Space-Valued Diagrams, Type-Theoretically (Extended Abstract),
Abstract: Topologists are sometimes interested in space-valued diagrams over a given
index category, but it is tricky to say what such a diagram even is if we look
for a notion that is stable under equivalence. The same happens in (homotopy)
type theory, where it is known only for special cases how one can define a type
of type-valued diagrams over a given index category. We offer several
constructions. We first show how to define homotopy coherent diagrams which
come with all higher coherence laws explicitly, with two variants that come
with assumption on the index category or on the type theory. Further, we
present a construction of diagrams over certain Reedy categories. As an
application, we add the degeneracies to the well-known construction of
semisimplicial types, yielding a construction of simplicial types up to any
given finite level. The current paper is only an extended abstract, and a full
version is to follow. In the full paper, we will show that the different
notions of diagrams are equivalent to each other and to the known notion of
Reedy fibrant diagrams whenever the statement makes sense. In the current
paper, we only sketch some core ideas of the proofs.