Related papers: The category of cellular resolutions
This paper studies the existence of model category structures on algebras and modules over operads in monoidal model categories.
A neural code on $ n $ neurons is a collection of subsets of the set $ [n]=\{1,2,\dots,n\} $. Curto et al. \cite{curto2013neural} associated a ring $\mathcal{R}_{\mathcal{C}}$ (neural ring) to a neural code $\mathcal{C}$. A special class of…
We study a family of monomial ideals, called block diagonal matching field ideals, which arise as monomial Gr\"obner degenerations of determinantal ideals. Our focus is on the minimal free resolutions of these ideals and all of their…
Whole-cell computational models aim to predict cellular phenotypes from genotype by representing the entire genome, the structure and concentration of each molecular species, each molecular interaction, and the extracellular environment.…
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…
In the theory of crossed modules, considering arbitrary self-actions instead of conjugation allows for the extension of the concept of crossed modules and thus the notion of generalized crossed module emerges. In this paper we give a…
In \cite{CompTheo} we studied the indeterminacy of the value of a derived functor at an object using different definitions of a derived functor and different types of fibrant replacement. In the present work we focus on derived or homotopy…
Consider a push-out diagram of spaces C <-- A --> B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy…
Let (R, m) be a local commutative ring. Suppose that m is principal and that m^2 = 0. We give a complete description of the cellular lattice of perfect chain complexes of modules over this ring.
Cellular categories are a generalization of cellular algebras, which include a number of important categories such as (affine)Temperley-Lieb categories, Brauer diagram categories, partition categories, the categories of invariant tensors…
Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper, we continue to develop theory in this area by proving a…
This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of $n$-nerve and Pelissier's thesis. If $M$ is a tractable left proper cartesian model category, we construct a…
After explaining the importance of model categories in abstract homotopy theory, we provide concrete examples demonstrating that various categories of manifolds do not have all finite colimits, and hence cannot be model categories. We then…
We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…
In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex…
Let $R$ be a commutative ring. We show that pure injective resolutions and pure projective resolutions can be constructed for unbounded complexes of $R$-modules. We use these to obtain a closed symmetric monoidal structure on the unbounded…
In this paper we will define notion homotopy of morphisms of crossed modules of Lie algebras. Then we construct a groupoid structure of Lie crossed module morphisms and their homotopies.
This paper is concerned with the relationships between two concepts, vanishing of cohomology groups and the structure of free resolutions. In particular, we study the connection between vanishing theorems for the local cohomology of…
The Godement cosimplicial resolution is available for a wide range of categories of sheaves. In this paper we investigate under which conditions of the Grothendieck site and the category of coefficients it can be used to obtain fibrant…
We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the $G$-comodules…