Related papers: Functor is to Lens as Applicative is to Biplate: I…
In this paper, we propose an algebraic approach to determine whether two non-isomorphic caterpillar trees can have the same symmetric function generalization of the chromatic polynomial. On the set of all composition on integers, we…
We introduce a new class of extensions of terms that consists in navigation strategies and insertion of contexts. We introduce an operation of combination on this class which is associative, admits a neutral element and so that each…
We introduce an extension of interpolation theory to more than two spaces by employing a functional parameter, while retaining a fully functorial and systematic framework. This approach allows for the construction of generalized…
Modular functors are traditionally defined as systems of projective representations of mapping class groups of surfaces that are compatible with gluing. They can formally be described as modular algebras over central extensions of the…
We introduce a notion of signature whose sorts form a direct category, and study computads for such signatures. Algebras for such a signature are presheaves with an interpretation of every function symbol of the signature, and we describe…
For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion…
A generally intelligent learner should generalize to more complex tasks than it has previously encountered, but the two common paradigms in machine learning -- either training a separate learner per task or training a single learner for all…
In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD --> D, which we call simple functor. For…
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…
Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual…
We present module theory and linear maps as a powerful generalised and computationally efficient framework for the relational data model, which underpins today's relational database systems. Based on universal constructions of modules we…
For a strongly connected category $\mathcal C$ with pair-wise coproducts, we introduce a cosimplicial object, which serves as a sort of resolution for computing higher derived functors of ${\sf lim} : \mathrm{Ab}^{\mathcal C}\to…
This paper contains analysis of concept of a class within different object-oriented knowledge representation models. The main attention is paid to structure of the class and its efficiency in the context of data storage, using…
MultiGrain is a network architecture producing compact vector representations that are suited both for image classification and particular object retrieval. It builds on a standard classification trunk. The top of the network produces an…
The main objective of the present paper is to present a version of the Tannaka-Krein type reconstruction Theorems: If $F:B\to C$ is an exact faithful monoidal functor of tensor categories, one would like to realize $B$ as category of…
We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal…
We describe new structure on the Goodwillie derivatives of a functor, and we show how the full Taylor tower of the functor can be recovered from this structure. This new structure takes the form of a coalgebra over a certain comonad which…
Let $B$ be a bialgebra, and $A$ a left $B$-comodule algebra in a braided monoidal category $\Cc$, and assume that $A$ is also a coalgebra, with a not-necessarily associative or unital left $B$-action. Then we can define a right $A$-action…
We prove a general congruence result for bisimilarity in higher-order languages, which generalises previous work to languages specified by a labelled transition system in which programs may occur as labels, and which may rely on operations…
Compositional generalization is the ability to generalize systematically to a new data distribution by combining known components. Although humans seem to have a great ability to generalize compositionally, state-of-the-art neural models…