Related papers: For Generalised Algebraic Theories, Two Sorts Are …
We present generalized algebraic theories corresponding to slightly modified versions of two of the type theories in our paper Type Theory with Explicit Universe Polymorphism. We first present a generalized algebraic theory for categories…
Algebraic theories with dependency between sorts form the structural core of Martin-L\"of type theory and similar systems. Their denotational semantics are typically studied using categorical techniques; many different categorical…
Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic descriptions of…
In functional programming languages, generalized algebraic data types (GADTs) are very useful as the unnecessary pattern matching over them can be ruled out by the failure of unification of type arguments. In dependent type systems, this is…
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of…
We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed…
We introduce a general axiomatic framework for algebras with triangular decomposition, which allows for a systematic study of the Bernstein-Gelfand-Gelfand Category $\mathcal{O}$. The framework is stated via three relatively simple axioms;…
Two novel descriptions of weak {\omega}-categories have been recently proposed, using type-theoretic ideas. The first one is the dependent type theory CaTT whose models are {\omega}-categories. The second is a recursive description of a…
We show that contrary to appearances, Multimodal Type Theory (MTT) over a 2-category M can be interpreted in any M-shaped diagram of categories having, and functors preserving, M-sized limits, without the need for extra left adjoints. This…
We intend to investigate the metalogical property of 'omitting types' for a wide variety of quantifier logics (that can also be seen as multimodal logics upon identifying existential quantifiers with modalities syntactically and…
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of…
Categories and categorical structures are increasingly recognized as useful abstractions for modeling in science and engineering. To uniformly implement category-theoretic mathematical models in software, we introduce GATlab, a…
Recursive algebraic data types (term algebras, ADTs) are one of the most well-studied theories in logic, and find application in contexts including functional programming, modelling languages, proof assistants, and verification. At this…
In various subjects including mathematics, one can hope to use mathematical thinking well when the right kinds of algebraic structure to consider can be discovered or spotted. Therefore, it would help to understand kinds of algebraic…
We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between ${\mathbb Z}$-graded algebras. As our main application of this theorem, we…
Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how…
We define a general class of dependent type theories, encompassing Martin-L\"of's intuitionistic type theories and variants and extensions. The primary aim is pragmatic: to unify and organise their study, allowing results and constructions…
We define and develop two-level type theory (2LTT), a version of Martin-L\"of type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is…
We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin-L\"{o}f type theory, two-level type theory and cubical type theory. We establish basic results in the semantics of type theory:…
We consider type inference for guarded recursive data types (GRDTs) -- a recent generalization of algebraic data types. We reduce type inference for GRDTs to unification under a mixed prefix. Thus, we obtain efficient type inference.…