We introduce a definition and framework for internal topological symmetries in quantum field
theory, including “noninvertible symmetries” and “categorical symmetries”. We outline a …

We prove the conjecture that any Grothendieck $(\infty, 1) $-topos can be presented by a
Quillen model category that interprets homotopy type theory with strict univalent universes …

We introduce a general theory of quantitative and metric rewriting systems, namely systems
with a rewriting relation enriched over quantales modelling abstract quantities. We develop …

To set the scene, we begin here by reviewing basics of homotopy theory via model category
theory [Quillen (1967)](review in [Hovey (1999)][Hirschhorn (2003)][Lurie (2009a), A. 2]) and …

Let $ X $ be a quasicompact quasiseparated scheme. Write $\operatorname {Gal}(X) $ for
the category whose objects are geometric points of $ X $ and whose morphisms are …

Reasoning about weak higher categorical structures constitutes a challenging task, even to
the experts. One principal reason is that the language of set theory is not invariant under the …

We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left
fibrations of simplicial spaces and studying their associated model structure, the covariant …

How do spaces in physics emerge from pregeometric discrete building blocks governed by
computational rules? To address this question we investigate non-deterministic rewriting …

The idea for this book was conceptualized during a research visit to Macquarie University in
the summer of 2017. At first, I was interested in collecting exotic examples of model …

We develop the theory of relative monads and relative adjunctions in a virtual equipment,
extending the theory of monads and adjunctions in a 2-category. The theory of relative …