Related papers: d-Complete posets: local structural axioms, proper…
We introduce Lipschitz functions on a finite partially ordered set $P$ and study the associated Lipschitz polytope $L(P)$. The geometry of $L(P)$ can be described in terms of descent-compatible permutations and permutation statistics that…
We show that all extended functorial field theories, both topological and nontopological, are local. We define the smooth (infinity,d)-category of bordisms with geometric data, such as Riemannian metrics or geometric string structures, and…
We consider the poset of vector partitions of $[n]$ into $s$ components, denoted $\Pi_{n,s}$, which was first defined by Stanley in 1978. In 1986, Sagan showed that this poset is CL-shellable, and hence has the homotopy type of a wedge of…
A poset can be regarded as a category in which there is at most one morphism between objects, and such that at most one of Hom(c,c') and Hom(c',c) is nonempty for c not equal to c'. If we keep in place the latter axiom but allow for more…
Duchamp--Hivert--Thibon introduced the construction of a right $H_n(0)$-module, denoted as $M_P$, for any partial order $P$ on the set $[n]$. This module is defined by specifying a suitable action of $H_n(0)$ on the set of linear extensions…
This paper is concerned with the taxonomy of finitely complete categories, based on 'matrix properties' - these are a particular type of exactness properties that can be represented by integer matrices. In particular, the main result of the…
In this article we introduce the $m$-cover poset of an arbitrary bounded poset $\mathcal{P}$, which is a certain subposet of the $m$-fold direct product of $\mathcal{P}$ with itself. Its ground set consists of multichains of $\mathcal{P}$…
Directed Algebraic Topology is beginning to emerge from various applications. The basic structure we shall use for such a theory, a 'd-space', is a topological space equipped with a family of 'directed paths', closed under some operations.…
We introduce partially ordered sets (posets) with an additional structure given by a collection of vector subspaces of an algebra $A$. We call them algebraically equipped posets. Some particular cases of these, are generalized equipped…
We continue studying net bundles over partially ordered sets (posets), defined as the analogues of ordinary fibre bundles. To this end, we analyze the connection between homotopy, net homology and net cohomology of a poset, giving versions…
We show that the category of chain-complete posets is nearly locally presentable, we give a characterization of strong epimorphisms, and we offer an alternative proof of co-wellpoweredness.
We introduce structured decompositions, category-theoretic structures which simultaneously generalize notions from graph theory (including treewidth, layered treewidth, co-treewidth, graph decomposition width, tree independence number,…
Lurie's representability theorem gives necessary and sufficient conditions for a functor to be an almost finitely presented derived geometric stack. We establish several variants of Lurie's theorem, making the hypotheses easier to verify…
We investigate Grothendieck topologies (in the sense of sheaf theory) on a poset $\P$ that are generated by some subset of $\P$. We show that such Grothendieck topologies exhaust all possibilities if and only if $\P$ is Artinian. If $\P$ is…
By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for any abelian category A, a functor between the categories of complexes of diagrams over X and Y with…
We define the covering gonality and separable covering gonality of varieties over arbitrary fields, generalizing the definition given by Bastianelli-de Poi-Ein-Lazarsfeld-Ullery for complex varieties. We show that over an arbitrary field a…
In this paper, we give a new completion for quasi-uniform spaces which generalizes the completion theories of Doitchinov [8] and Stoltenberg [20]. The presented completion theory is very well-behaved and extends the completion theory of…
We introduce a novel combinatorial structure called pointed building sets, which can be viewed as families of lattices equipped with compatibility relations. To each pointed building set $\mathsf{B}$, we associate a complete lattice…
Examples of exact categories in representation theory are given by the category of Delta-filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or…
These are expanded notes from some talks given during the fall 2002, about ``homotopical algebraic geometry'' (HAG) with special emphasis on its applications to ``derived algebraic geometry'' (DAG) and ``derived deformation theory''. We use…