Take M , a finite-dimensional differentiable manifold, and f : M → R a smooth function. Such a function f is called a Morse function if it has no degenerate critical points. Morse theory allows us to connect the topology, in particular the homotopy type, of M with the behavior of f on M . In the following sections, we will state and prove two important theorems in Morse theory. Using these two ...

We construct a small realization as flow of every precubical set (modeling for example a process algebra). The realization is small in the sense that the construction does not make use of any cofibrant replacement functor and of any transfinite construction. In particular, if the precubical set is finite, then the corresponding flow has a finite globular decomposition. Two applications are give...

We introduce the persistent homotopy type distance dHT to compare real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of dHT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is inte...

Homotopy Type Theory may be seen as an internal language for the ∞category of weak ∞-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak ∞-groupoids as a new foundation for mathematics called the Univalent Foundations of Mathematics. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of)...

We show that Martin Hyland's effective topos can be exhibited as the homotopy category of a path category $\mathbb{EFF}$. Path categories are categories of fibrant objects in the sense of Brown satisfying two additional properties and as such provide a context in which one can interpret many notions from homotopy theory and Homotopy Type Theory. Within the path category $\mathbb{EFF}$ one can i...