Related papers: Univalence and Constructive Identity
We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy…
In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in…
By a theorem of Chevalley the image of a morphism of varieties is a constructible set. The algebraic version of this fact is usually stated as a result on "extension of specializations" or "lifting of prime ideals". We present a difference…
We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a homotopy-initial algebra. This notion is defined by a…
We present Nonstandard Analysis by three axioms: the {\em Extension, Transfer and Saturation Principles} in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to…
We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More…
We show that discrete and classical homotopy theories are equivalent after localizing at n-equivalences for any non-negative integer n. By constructing an explicit homotopy inverse to the graph nerve functor associating an n-fibrant cubical…
We study the coherence and conservativity of extensions of dependent type theories by additional strict equalities. By considering notions of congruences and quotients of models of type theory, we reconstruct Hofmann's proof of the…
This piece threads substructurality and modality into a negation that activates the downside of equivalence and identity in a fibrational framework. The piece is a working through of negation and contradiction as type-theoretic/categorial…
We consider the problem of characterizing isomorphisms of types, or, equivalently, constructive cardinality of sets, in the simultaneous presence of disjoint unions, Cartesian products, and exponentials. Mostly relying on results about…
A classical problem of statistical inference is the valid specification of a model that can account for the statistical dependencies between observations when the true structure is dense, intractable, or unknown. To address this problem, a…
We present a simple alternative viewpoint on Hodge-Newton indecomposability, illustrating its explanatory value through a uniform proof of a combinatorial identity arising from affine Deligne-Lusztig varieties with finite Coxeter part.
The idea of this approach towards proving the consistency of Quine's New Foundations set theory is to go in a completely untyped manner. So no contemplation about types is utilized here. All conceptualization pivots around proving a handful…
Cutting-edge image generation has been praised for producing high-quality images, suggesting a ubiquitous future in a variety of applications. However, initial studies have pointed to the potential for harm due to predictive bias,…
Homotopy type theory (HoTT) can be seen as a generalisation of structural set theory, in the sense that 0-types represent structural sets within the more general notion of types. For material set theory, we also have concrete models as…
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…
A multiset consists of elements, but the notion of a multiset is distinguished from that of a set by carrying information of how many times each element occurs in a given multiset. In this work we will investigate the notion of iterative…
Unveiling individuals' preferences for connecting with similar others (choice homophily) beyond the structural factors determining the pool of opportunities, is a challenging task. Here, we introduce a robust methodology for quantifying and…
The two pillars of Algebraic topology - Homology and homotopy theory rely on the availability of basic building blocks called cells. Cells take the form of simplexes, and have properties such as faces, sub-cells, convexity and…
From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The procedure is similar to an ultrapower construction of an internal set from a sequence of subsets of the real line, but now the individual entities…