A Model Structure on the Category of -Categories with Strict Morphisms.

Given a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.

The flat stable module category of a coherent ring

In this paper, we give a new definition of partial Higher Dimension Automata using lax functors. This definition is simpler and more natural from a categorical point of view, but also matches more clearly the intuition that pHDA are Higher Dimensional Automata with some missing faces. We then focus on trees. Originally, for example in transition systems, trees are defined as those systems that have a unique path property. To understand what kind of unique property is needed in pHDA, we start by looking at trees as colimits of paths. This definition tells us that trees are exactly the pHDA with the unique path property modulo a notion of homotopy, and without any shortcuts. This property allows us to prove two interesting characterisations of trees: trees are exactly those pHDA that are an unfolding of another pHDA; and trees are exactly the cofibrant objects, much as in the language of Quillen’s model structure. In particular, this last characterisation gives the premisses of a new understanding of concurrency theory using homotopy theory.

A Quillen model for classical Morita theory and a tensor categorification of the Brauer group

We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, i.e., the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, i.e., the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categories yield Quillen equivalences of model categories of algebras over operads. Applications of these results include enriched categories, colored operads, prefactorization algebras, and commutative symmetric ring spectra.

Cosimplicial versus DG-rings: a version of the Dold–Kan correspondence

We first study commutative, pointed monoids providing basic definitions and results in a manner similar commutative ring theory. Included are results on chain conditions, primary decomposition as well as normalization for a special class of monoids which lead to a study monoid schemes, divisors, Picard groups and class groups. It is shown that the normalization of a monoid need not be a monoid, but possibly a monoid scheme. After giving the definition of, and basic results for, $A$-sets, we classify projective $A$-sets and show they are completely determine by their rank. Subsequently, for a monoid $A$, we compute $K_0$ and $K_1$ and prove the Devissage Theorem for $G_0$. With the definition of short exact sequence for $A$-sets in hand, we describe the set $Ext(X,Y)$ of extensions for $A$-sets $X,Y$ and classify the set of square-zero extensions of a monoid $A$ by an $A$-set $X$ using the Hochschild cosimplicial set. We also examine the projective model structure on simplicial $A$-sets showcasing the difficulties involved in computing homotopy groups as well as determining the derived category for a monoid. The author defines the category $\operatorname{Da}(\mathcal{C})$ of double-arrow complexes for a class of non-abelian categories $\mathcal{C}$ and, in the case of $A$-sets, shows an adjunction with the category of simplicial $A$-sets.

Chiral stationary phases based on chicken alpha 1-acid glycoprotein (cAGP) have been used for enantioseparations of various compounds. However, the chiral binding sites and mechanism have not been clarified yet. Based on chromatographic properties of native and W26-modified cAGP columns and docking simulations of studied compounds into the generated model structure of cAGP, the chiral binding sites were located on cAGP and the chiral binding mechanism was discussed. On cAGP, there existed a binding cavity lined with H25, W26, Y47, R128, T129, D161 and E168, which contribute electrostatic or hydrogen bonding interactions. Benzoin and chlorpheniramine enantiomers interacted with cAGP at almost the same sites a little away from W26, while propranolol enantiomers docked, slightly shifting toward H25 and W26. Furthermore, in addition to hydrophobic interactions, ionic interactions between amino groups of chlorpheniramine enantiomers and a carboxyl group of D161 or E168 played an important role in the chiral recognition, while hydrophobic interactions and hydrogen bonding interactions worked for the chiral recognition of benzoin and propranolol enantiomers.

Ab initio study of charged states of H in amorphous SiO2