Related papers: Higher Structures in Homotopy Type Theory
We study the realization problem of finite groups as the group of homotopy classes of self-homotopy equivalences of finite spaces. Let $G$ be a finite group. Using an infinite family of pairwise non weakly homotopic asymmetric spaces we…
We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class…
An appropriate framework is put forward for the construction of $\lambda$-models with $\infty$-groupoid structure, which we call \textit{homotopic $\lambda$-models}, through the use of an $\infty$-category with cartesian closure and enough…
This paper studies the homotopy theory of algebras and homotopy algebras over an operad. It provides an exhaustive description of their higher homotopical properties using the more general notion of morphisms called infinity-morphisms. The…
When working in Homotopy Type Theory and Univalent Foundations, the traditional role of the category of sets, Set, is replaced by the category hSet of homotopy sets (h-sets); types with h-propositional identity types. Many of the properties…
Higher structures - infinity algebras and other objects up to homotopy, categorified algebras, `oidified' concepts, operads, higher categories, higher Lie theory, higher gauge theory... - are currently intensively investigated in…
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, Shulman) solves this by considering…
The aim of homotopy theory in topology is to simplify, after continuous deformation, continuous maps between topological spaces. What prevents this from happening are homotopy invariants. This raises quantitative questions: $\bullet$ Is the…
We construct model category structures on various types of (marked) *-categories. These structures are used to present the infinity categories of (marked) *-categories obtained by inverting (marked) unitary equivalences. We use this…
In this text we expose basic cases of some fundamental ideas and methods of topology. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal…
This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often…
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…
We build free, bigraded bidifferential algebra models for the forms on a complex manifold, with respect to a strong notion of quasi-isomorphism and compatible with the conjugation symmetry. This answers a question of Sullivan. The resulting…
Homotopy Type Theory with a univalent universe $\,\mathcal{U}_0$ is interpreted at the strength of finite order arithmetic. We eliminate Grothendieck universes, avoid the axiom of replacement, and bound all uses of separation.
We observe that the notion of two sets being equal up to finitely many elements is a homotopy equivalence relation in a model category, and suggest a homotopy-invariant variant of Generalised Continuum Hypothesis about which more can be…
Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to…
Homotopy Type Theory may be seen as an internal language for the $\infty$-category of weak $\infty$-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak $\infty$-groupoids as a new foundation…
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…
Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened "homotopy type theory". In this…
Polynomial functors are a categorical generalization of the usual notion of polynomial, which has found many applications in higher categories and type theory: those are generated by polynomials consisting a set of monomials built from sets…