Jonas Höfer

Pamela Fleischmann, Jonas Höfer, Annika Huch et al.

In 1991 Hébrard introduced a factorization of words that turned out to be a powerful tool for the investigation of a word's scattered factors (also known as (scattered) subwords or subsequences). Based on this, first Karandikar and Schnoebelen introduced the notion of $k$-richness and later on Barker et al. the notion of $k$-universality. In 2022 Fleischmann et al. presented a generalization of the arch factorization by intersecting the arch factorization of a word and its reverse. While the authors merely used this factorization for the investigation of shortest absent scattered factors, in this work we investigate this new $α$-$β$-factorization as such. We characterize the famous Simon congruence of $k$-universal words in terms of $1$-universal words. Moreover, we apply these results to binary words. In this special case, we obtain a full characterization of the classes and calculate the index of the congruence. Lastly, we start investigating the ternary case, present a full list of possibilities for $αβα$-factors, and characterize their congruence.

Thierry Coquand, Jonas Höfer, Christian Sattler

There have recently been several developments in synthetic mathematics using extensions of dependent type theory with univalence and higher inductive types: simplicial homotopy type theory, synthetic algebraic geometry and synthetic Stone duality. We provide a foundation of higher sheaf models of type theory in a constructive metatheory and, in particular, build constructive models of these formal systems.

Evan Cavallo, Jonas Höfer

It is a well-known theorem of homotopy type theory, originally due to Voevodsky, that function extensionality holds inside any univalent universe. We consider a weaker variant of the univalence axiom, asserting that the wild category formed by the universe is univalent, which we call categorical univalence. We show that categorical univalence does not imply function extensionality by an analysis of Von Glehn's polynomial model construction, which produces models of Martin-Löf type theory that always refute function extensionality. We find in particular that when the base model has a univalent universe, its polynomial model has a universe that is categorically univalent but lacks function extensionality.

Pamela Fleischmann, Tero Harju, Lukas Haschke et al.

The shuffle product \(u\shuffle v\) of two words \(u\) and \(v\) is the set of all words which can be obtained by interleaving \(u\) and \(v\). Motivated by the paper \emph{The Shuffle Product: New Research Directions} by Restivo (2015) we investigate a special case of the shuffle product. In this work we consider the shuffle of a word with itself called the \emph{self shuffle} or \emph{shuffle square}, showing first that the self shuffle language and the shuffle of the language are in general different sets. We prove that the language of all words arising as a self shuffle of some word is context sensitive but not context free. Furthermore, we show that the self shuffle \(w \shuffle w\) uniquely determines \(w\).