Related papers: Polynomial Universes in Homotopy Type Theory
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…
Presheaf models of dependent type theory have been successfully applied to model HoTT, parametricity, and directed, guarded and nominal type theory. There has been considerable interest in internalizing aspects of these presheaf models,…
Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\infty$-)categories of spans (or correspondences). In this paper we study the…
For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture…
We give a type system in which the universe of types is closed by reflection into it of the logical relation defined externally by induction on the structure of types. This contribution is placed in the context of the search for a natural,…
We introduce judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and natural deduction as special kinds of…
We consider the natural generalization of the notion of the order of a phantom map from the topological setting to triangulated categories. When applied to the derived category of the category of countable flat modules over a countable…
Following the types-as-sets paradigm, we present a mechanized embedding of dependent function types with a hierarchy of universes into schematic first-order logic with equality, with axiom schemas of Tarski-Grothendieck set theory. We carry…
In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We show that the two definitions are equivalent…
We give a model of set theory based on multisets in homotopy type theory. The equality of the model is the identity type. The underlying type of iterative sets can be formulated in Martin-L\"of type theory, without Higher Inductive Types…
We prove a conjecture about the constructibility of coinductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence of inductive types, coinductive types are derivable.…
We study the category $\mathcal{F}(\mathfrak{S}_S,\mathcal{V})$ of functors from the category $\mathfrak{S}_S$, which is the category of elements of some presheaf $S$ on the category $\mathcal{V}^f$ of finite dimensional vector spaces, to…
This paper develops a version of dependent type theory in which isomorphism is handled through a direct generalization of the 1939 definitions of Bourbaki. More specifically we generalize the Bourbaki definition of structure from simple…
A well-known challenge in the semantics of programming languages is how to combine non-determinism and probability. At a technical level, the problem arises from the fact that there is a no distributive law between the powerset monad and…
Motivated by gauge theory, we develop a general framework for chain complex valued algebraic quantum field theories. Building upon our recent operadic approach to this subject, we show that the category of such theories carries a canonical…
We propose foundations for a synthetic theory of $(\infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of…
We extend the theory of distributive series of monads of \cite{EC1} by extending the definition to include an $\bN$-indexed collection of monads. Under certain conditions, distributive series of monads will have a colimit in the category of…
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can…
An elementary notion of homotopy can be introduced between arrows in a cartesian closed category $E$. The input is a finite-product-preserving endofunctor $\Pi_0$ with a natural transformation $p$ from the identity which is surjective on…
We give sufficient conditions which ensure that a functor of finite length from an additive category to finite-dimensional vector spaces has a projective resolution whose terms are finitely generated. For polynomial functors, we study also…