Related papers: Elementary equivalences and accessible functors

Researchers have recently suggested that models share common representations. In our work, we find numerous geometric similarities across the token embeddings of large language models. First, we find ``global'' similarities: token…

We introduce a category-theoreticabstraction of a syntax with auxiliary functions, called an admissiblemonad morphism. Relying on an abstract form of structural recursion,we then design generic tools to construct admissible monad…

In this paper we provide a detailed construction of an equivalence between the category of Lawvere theories and the category of relative monads on the obvious functor $Jf:F\rightarrow Sets$ where $F$ is the category with the set of objects…

We present a categorical formulation of the Hamiltonian renormalisation programme for quantum field theories, establishing a systematic bridge between functional and lattice renormalisation. To this end, we introduce two categories, $Seq$…

We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we…

In this note, we aim to establish a number of embeddings between various function spaces that are frequently considered in the theory of Fourier series. More specifically, we give sufficient conditions for the embeddings $\Phi V[h]\subseteq…

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable…

The categorical models of the differential lambda-calculus are additive categories because of the Leibniz rule which requires the summation of two expressions. This means that, as far as the differential lambda-calculus and differential…

Despite the recent successes of deep learning in natural language processing (NLP), there remains widespread usage of and demand for techniques that do not rely on machine learning. The advantage of these techniques is their…

We show that the principal types of the closed terms of the affine fragment of $\lambda$-calculus, with respect to a simple type discipline, are structurally isomorphic to their interpretations, as partial involutions, in a natural Geometry…

Many elliptic boundary value problems exhibit an interior regularity property, which can be exploited to construct local approximation spaces that converge exponentially within function spaces satisfying this property. These spaces can be…

Exponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows: the elementary conditions on a given functor translate into existence statements for certain adjoints that obey…

Both the Klein-Williams invariant $\ell_G(f)$ from \cite{KW2} and the generalized equivariant Lefschetz invariant $\lambda_G(f)$ from \cite{weber07} serve as complete obstructions to the fixed point problem in the equivariant setting. The…

We construct a functor from the category of graphs to the category of groups which is faithful and "almost" full, in the sense that it induces bijections of the Hom sets up to trivial homomorphisms and conjugation in the category of groups.…

This paper unifies problems and results related to (embedding) universal and homomorphism universal structures. On the one side we give a new combinatorial proof of the existence of universal objects for homomorphism defined classes of…

In this paper we scrutinize the so called Principle of Local Lorentz Invariance (\emph{PLLI}) that many authors claim to follow from the Equivalence Principle. Using rigourous mathematics we introduce in the General Theory of Relativity two…

We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms…

We define and study the notion of a locally bounded enriched category over a (locally bounded) symmetric monoidal closed category, generalizing the locally bounded ordinary categories of Freyd and Kelly. In addition to proving several…

The present work is devoted to an extension of the well-known Ehrling inequalities, which quantitatively characterize compact embeddings of function spaces, to more general operators. Firstly, a modified notion of continuity for linear…

Orbifold equivalence is a notion of symmetry that does not rely on group actions. Among other applications, it leads to surprising connections between hitherto unrelated singularities. While the concept can be defined in a very general…