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We show that every braided monoidal category arises as $\End(I)$ for a weak unit $I$ in an otherwise completely strict monoidal 2-category. This implies a version of Simpson's weak-unit conjecture in dimension 3, namely that one-object…
We uncover a close relationship between combinatorial and syntactic proofs for first-order logic (without equality). Whereas syntactic proofs are formalized in a deductive proof system based on inference rules, a combinatorial proof is a…
We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or `S-operads', and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad…
This paper continues the development of a simplicial theory of weak omega-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of…
We introduce a 3-dimensional categorical structure which we call intercategory. This is a kind of weak triple category with three kinds of arrows, three kinds of 2-dimensional cells and one kind of 3-dimensional cells. In one dimension, the…
The strict globular $\omega$-categories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) $\omega$-category $\C$ three homology theories. The first one is called the…
Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information.…
In [math.AT/9907138] we proved that strongly homotopy algebras are homotopy invariant concepts in the category of chain complexes. Our arguments were based on the fact that strongly homotopy algebras are algebras over minimal cofibrant…
A compatible associative algebra is a vector space endowed with two associative multiplication operations that satisfy a natural compatibility condition. In this paper, we investigate and classify compatible pairs of associative algebras of…
It is known that the so-called monadic decomposition, applied to the adjunction connecting the category of bialgebras to the category of vector spaces via the tensor and the primitive functors, returns the usual adjunction between…
Cover's function counting theorem is a milestone in the theory of artificial neural networks. It provides an answer to the fundamental question of determining how many binary assignments (dichotomies) of $p$ points in $n$ dimensions can be…
Graph classification plays an important role is data mining, and various methods have been developed recently for classifying graphs. In this paper, we propose a novel method for graph classification that is based on homotopy equivalence of…
This paper develops a methodology for representing machine learning models as models of formal theories, grounded in the perspective that machine learning models are a form of database and that databases are models of theories in coherent…
We introduce and analyze parallelizable algorithms to compress and accurately reconstruct finite simplicial complexes that have non-trivial automorphisms. The compressed data -- called a complex of groups -- amounts to a functor from (the…
Many theories of semantic interpretation use lambda-term manipulation to compositionally compute the meaning of a sentence. These theories are usually implemented in a language such as Prolog that can simulate lambda-term operations with…
Collections of measures on compact metric spaces form a model category ("data complexes"), whose morphisms are marginalization integrals. The fibrant objects in this category represent collections of measures in which there is a measure on…
We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching…
The assembly index of assembly theory quantifies the minimal number of composition steps required to construct an object from elementary components. The study proves that the decision version of the assembly index problem is NP-complete,…
In this paper, a novel learning paradigm is presented to automatically identify groups of informative and correlated features from very high dimensions. Specifically, we explicitly incorporate correlation measures as constraints and then…
Contraction analysis considers the distance between two adjacent trajectories. If this distance is contracting, then trajectories have the same long-term behavior. The main advantage of this analysis is that it is independent of the…