Related papers: Categoricity for Patterns of Order 2
Every definably complete expansion of an ordered field satisfies an analogue of the Baire Category Theorem.
We propose a model-theoretic structure for Shimura varieties and give necessary and sufficient conditions to obtain categoricity. We show that these conditions are directly related to important conjectures in number theory coming from…
We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different "sorts." We prove a rigidification result for simplicial algebras over these theories, showing…
We establish two versions of a central theorem, the Family Colimit Theorem, for the coarse coherence property of metric spaces. This is a coarse geometric property and so is well-defined for finitely generated groups with word metrics. It…
We provide a new proof of a important theorem in the Lagrangian formalism about necessary and sufficient conditions for a second-order variational system of equations to follow from a first-order Lagrangian.
We give a categorical account of Arrow's theorem, a seminal result in social choice theory.
We introduce an equivalence relation on the global class of morphisms of a category that extends several classical notions of equivalence in mathematics. We show that the standard group-action equivalence is a special case of our framework.…
This paper presents the proof of the coherence theorem for Ann-categories whose set of axioms and original basic properties were given in [9]. Let $$\A=(\A,{\Ah},c,(0,g,d),a,(1,l,r),{\Lh},{\Rh})$$ be an Ann-category. The coherence theorem…
Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and…
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…
A new approach to the construction of general persistent polyhierarchical classifications is proposed. It is based on implicit description of category polyhierarchy by a generating polyhierarchy of classification criteria. Similarly to…
Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops \textit{from the ground up} an abstract algebraic and qualitative notion of similarity based on the…
The purpose of this article is to present ideas towards obtaining a model category structure on the category of small strict n-categories, generalizing the one obtained by Thomason on ordinary categories. Following ideas of Grothendieck and…
We generalize several recognizability theorems for free single-sorted algebras to the field of many-sorted algebras and provide, in a uniform way and without using neither regular tree grammars nor tree automata, purely algebraic proofs of…
The concept of configuration was first introduced to give a characterization for the amenability of groups. Then the concept of two-sided configuration was suggested to provide normality to study the group structures more efficiently. It…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
Coherence phenomena appear in two different situations. In the context of category theory the term `coherence constraints' refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the…
An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…
Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this…
The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of $\mathcal{F}$-quotients in a…