Related papers: Neural category
Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship between network connectivity and encoded patterns is still…
In this paper, we introduced a generalization of the derived category, which is called the $n$-derived category and denoted by $\D_{n}(R)$, of a given ring $R$ for each $n\in\mathbb{N}\cup\{\infty\}$. The $n$-derived category of a ring is…
We extend the notion of graph homomorphism to cellularly embedded graphs (maps) by designing operations on vertices and edges that respect the surface topology; we thus obtain the first definition of map homomorphism that preserves both the…
Let $A$ be a Noetherian ring and let $\mathcal{R} = \bigoplus_{n \geq 0}\mathcal{R}_n$ be a standard graded ring with $\mathcal{R}_0 = A$. We define a category $\mathfrak{A}(\mathcal{R})$ of graded $\mathcal{R}$-modules (not necessarily…
Although it seems counter-intuitive, categorical colours do not exist as external physical entities but are very much the product of our brains. Our cortical machinery segments the world and associate objects to specific colour terms, which…
For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the category are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms are smooth $d$-dimensional equivariant cobordisms. We identify…
Sensory stimuli are usually composed of different features (the what) appearing at irregular times (the when). Neural responses often use spike patterns to represent sensory information. The what is hypothesized to be encoded in the…
This paper proposes a formal cognitive framework for problem solving based on category theory. We introduce cognitive categories, which are categories with exactly one morphism between any two objects. Objects in these categories are…
We show that the homotopy theory of strict 2-categories embeds in that of $(\infty,2)$-categories in the form of 2-precomplicial sets. More precisely, we construct a nerve-categorification adjunction that is a Quillen pair between Lack's…
We discuss the relationship between (co)homology groups and categorical diagonalization. We consider the category of chain complexes in the category of finitely generated free modules on a commutative ring. For a fixed chain complex with…
Cellular resolutions is a well studied topic on the level of single resolutions and certain specific families of cellular resolutions. One question coming out of the work on families is to understand the structure of cellular resolutions…
This paper presents a batch classifier that has been improved from the earlier version and fixed a mistake in the earlier paper. Two important changes have been made. Each category is represented by a classifier, where each classifier…
Neural networks can be thought of as applying a transformation to an input dataset. The way in which they change the topology of such a dataset often holds practical significance for many tasks, particularly those demanding non-homeomorphic…
We introduce a notion of globular multicategory with homomorphism types. These structures arise when organizing collections of "higher category-like" objects such as type theories with identity types. We show how these globular…
Morphisms between (formal) contexts are certain pairs of maps, one between objects and one between attributes of the contexts in question. We study several classes of such morphisms and the connections between them. Among other things, we…
A $\mathcal{C}$-set is a functor from the category $\mathcal{C}$ to the category of finite sets and functions. The category of $\mathcal{C}$-sets, $\mathcal{C} - \operatorname*{set}$, is defined as the category whose objects are…
In this paper we develop a novel mathematical formalism for the modeling of neural information networks endowed with additional structure in the form of assignments of resources, either computational or metabolic or informational. The…
In this paper we present the notion of a von Neumann regular $\mathcal{C}^{\infty}-$ring, we prove some results about them and we describe some of their properties. We prove, using two different methods, that the category of von Neumann…
The groupoid of finite sets has a "canonical" structure of a symmetric 2-rig with the sum and product respectively given by the coproduct and product of sets. This 2-rig $\widehat{\mathbb{F}\mathbb{S} et}$ is just one of the many…
In this paper we obtain several model structures on {\bf DblCat}, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double…