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We construct the quadratic analogue of the boson Fock functor. While in the first order case all contractions on the 1--particle space can be second quantized, the semigroup of contractions that admit a quadratic second quantization is much…
We thoroughly treat several familiar and less familiar definitions and results concerning categories, functors and distributors enriched in a base quantaloid Q. In analogy with V-category theory we discuss such things as adjoint functors,…
Embeddings mapping high-dimensional discrete input to lower-dimensional continuous vector spaces have been widely adopted in machine learning applications as a way to capture domain semantics. Interviewing 13 embedding users across…
It is known that there is a weak-equivalence between the geometric realization of a simplicially enriched small category and its cofibrant replacement [12]. In this paper, we show that when only small categories are considered there exists…
The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for…
Modern deep learning-based recommendation systems exploit hundreds to thousands of different categorical features, each with millions of different categories ranging from clicks to posts. To respect the natural diversity within the…
In this paper, we classify the function space of all real-valued functions on R denoted as F(R, R) into 28 distinct blocks. Each block contains elements that share common features in terms of the cardinality of their sets of continuity and…
Embedders play a central role in machine learning, projecting any object into numerical representations that can, in turn, be leveraged to perform various downstream tasks. The evaluation of embedding models typically depends on…
We study a new notion of reduction between structures called enumerable functors related to the recently investigated notion of computable functors. Our main result shows that enumerable functors and effective interpretability with the…
For conformal field theories in arbitrary dimensions, we introduce a method to derive the conformal blocks corresponding to the exchange of a traceless symmetric tensor appearing in four point functions of operators with spin. Using the…
By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for any abelian category A, a functor between the categories of complexes of diagrams over X and Y with…
For a semisimple multiring category with left duals, we prove that the unit object is simple if and only if the tensor functors by any non-zero algebra are separable (resp. faithful, resp. Maschke, resp. dual Maschke, resp. conservative).…
Abstractions of causal models allow for the coarsening of models such that relations of cause and effect are preserved. Whereas abstractions focus on the relation between two models, in this paper we study a framework for causal embeddings…
The issues of quintessence and cosmic acceleration can be discussed in the framework of higher order theories of gravity. We can define effective pressure and energy density directly connected to the Ricci scalar of curvature of a generic…
Conceptual spaces are geometric representations of conceptual knowledge, in which entities correspond to points, natural properties correspond to convex regions, and the dimensions of the space correspond to salient features. While…
It is shown that certain transformations on quiver-dimension vector pairs induce isomorphisms on the corresponding moduli spaces of quiver representations and map a stable dimension vector to a stable dimension vector. This result combined…
The sequence space of all real-valued sequences, denoted $Seq(\mathbb{R})$, is typically investigated through the lens of infinite-dimensional vector spaces, utilizing Banach space norms or Schauder bases. This work proposes a…
Recent work in set theory indicates that there are many different notions of 'set', each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to give one class theory that allows for a…
We construct a symmetric monoidal closed category of polynomial endofunctors (as objects) and simulation cells (as morphisms). This structure is defined using universal properties without reference to representing polynomial diagrams and is…
A classical approach to investigate a closed projective scheme $W$ consists of considering a general hyperplane section of $W$, which inherits many properties of $W$. The inverse problem that consists in finding a scheme $W$ starting from a…