Related papers: Realisability problem in arrow categories

We show a first rectification result for homotopy chain coalgebras over a field. On the one hand, we consider the $\infty$-category obtained by localizing differential graded coalgebras over an operad with respect to quasi-isomorphisms; on…

We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the o-minimal LS-category of definable groups. Along…

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$…

We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…

We study the question when for a given *-algebra $\mathcal{A}$ a sequence of cones $C_n\in M_n(\mathcal{A})$ can be realized as cones of positive operators in a faithful *-representation of $\mathcal{A}$ on a Hilbert space. A…

For every action $\phi\in\text{Hom}(G,\text{Aut}_k(K))$ of a group $G$ on a commutative ring $K$ we introduce two abelian monoids. The monoid $\text{Cliff}_k(\phi)$ consists of equivalent classes of $G$-graded Clifford system extensions of…

The universal (co)acting bi/Hopf algebras introduced by Yu. I. Manin, M. Sweedler and D. Tambara, the universal Hopf algebra of a given (co)module structure, as well as the universal group of a grading, introduced by J. Patera and H.…

We show that Wall's D(2) problem, the Realization problem and the Relation Gap problem could all be solved if it could be shown that the deficiency of a certain group is, as intuition would suggest, less than -1. Note the paper has been…

Many computational problems can be modelled as the class of all finite structures $\mathbb A$ that satisfy a fixed first-order sentence $\phi$ hereditarily, i.e., we require that every (induced) substructure of $\mathbb A$ satisfies $\phi$.…

We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\Gamma)$ for $\Gamma$ a finitely generated group with solvable word problem, $C^*(\Gamma)$ for $\Gamma$ a finitely…

Let $G=G_1 \times G_2$ be a finite group. We know that the second cohomology group $H^2(G,\mathbb C^\times)$ is isomorphic to $H^2(G_1,\mathbb C^\times) \times H^2(G_2,\mathbb C^\times) \times Hom(G_1/G_1' \otimes_\mathbb Z G_2/G_2',…

We construct a category $\OrdFor$ as an arboreal extension of $\Delta_{\mathrm{epi}}\subseteq\Delta$, whose morphisms are ordered forests composed by grafting. We define a full functor $\pi\colon \OrdFor\to\Delta_{\mathrm{epi}}^{op}$…

The category ${\rm Rel}(\mathcal{C})$ may be formed for any category $\mathcal{C}$ with finite limits using the same objects as $\mathcal{C}$ but whose morphisms from $X$ to $Y$ are binary relations in $\mathcal{C}$, that is, subobjects of…

Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to…

A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of…

Let $I$ be an infinite set, $\{G_i:i\in I\}$ be a family of (topological) groups and $G=\prod_{i\in I} G_i$ be its direct product. For $J\subseteq I$, $p_{J}: G\to \prod_{j\in J} G_j$ denotes the projection. We say that a subgroup $H$ of…

A monoidal category has a natural isomorphism $\alpha_{A,B,C}\colon(A\otimes B)\otimes C\to A\times (B\otimes C)$ called the associator. In the case where the objects $(A\otimes B)\otimes C$ and $A\otimes(B\otimes C)$ are equal, it is…

Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…

Let $A$ be a unital simple separable exact C$^*$-algebra which is approximately divisible and of real rank zero. We prove that the set of positive elements in $A$ with a fixed non-compact Cuntz class has vanishing homotopy groups. Combined…