Related papers: Algebraic models of dependent type theory

In this paper, I establish the categorical structure necessary to interpret dependent inductive and coinductive types. It is well-known that dependent type theories \`a la Martin-L\"of can be interpreted using fibrations. Modern theorem…

This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative…

Native type systems are those in which type constructors are derived from term constructors, as well as the constructors of predicate logic and intuitionistic type theory. We present a method to construct native type systems for a broad…

We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a typed presentation hence there are no…

Following ideas of Lawvere and Linton we prove that classical varieties are precisely the exact categories with a varietal generator. This means a strong generator which is abstractly finite and regularly projective. An analogous…

Databases have been studied category-theoretically for decades. The database schema -- whose purpose is to arrange high-level conceptual entities -- is generally modeled as a category or sketch. The data itself, often called an instance, is…

We present a domain-specific type theory for constructions and proofs in category theory. The type theory axiomatizes notions of category, functor, profunctor and a generalized form of natural transformations. The type theory imposes an…

On a finite structure, the polymorphism invariant relations are exactly the primitively positively definable relations. On infinite structures, these two sets of relations are different in general. Infinitary primitively positively…

We define model structures on exact categories which we call exact model structures. We look at the relationship between these model structures and cotorsion pairs on the exact category. In particular, when the underlying category is weakly…

We prove that a large class of natural transformations (consisting roughly of those constructed via composition from the "functorial" or "base change" transformations) between two functors of the form $\cdots f^* g_* \cdots$ actually has…

AlphaZero learns to play go, chess and shogi at a superhuman level through self play given only the rules of the game. This raises the question of whether a similar thing could be done for mathematics -- a MathZero. MathZero would require a…

The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods…

Representation learning seeks meaningful sensory representations without supervision and can model aspects of human development. Although many neural networks empirically learn useful features, a principled account of what makes a…

We develop the necessary theory in computational algebraic geometry to place Bayesian networks into the realm of algebraic statistics. We present an algebra{statistics dictionary focused on statistical modeling. In particular, we link the…

We define a notion of equivalence between algebraic dependent type theories which we call Morita equivalence. This notion has a simple syntactic description and an equivalent description in terms of models of the theories. The category of…

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…

Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any…

We assemble polynomials in a locally cartesian closed category into a tricategory, allowing us to define the notion of a polynomial pseudomonad and polynomial pseudoalgebra. Working in the context of natural models of type theory, we prove…

We introduce a notion of (co)presheaf on a lax double functor $X$, which we generally call an instance. In the terminology of double-categorical logic, a lax double functor valued in sets, possibly preserving finite products, is called a…

We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions,…