arXiv subjects

Sanjeevi Krishnan

Publications and source records attributed to Sanjeevi Krishnan.

12 recordsLinked to original sources

arXiv2025

A convenient category of cubes

We claim that the cube category whose morphisms are the interval-preserving monotone functions between finite Boolean lattices is a convenient general-purpose site for cubical sets. This category is the largest possible concrete Eilenberg-Zilber variant excluding the reversals and diagonals. The category admits as monoidal generators all functions between the singleton and two-element ordinals and all monotone surjections from finite Boolean lattices to the two-element ordinal. Consequently, morphisms in the minimal symmetric monoidal variant of the cube category containing coconnections of one kind can be characterized as the interval-preserving semilattice homomorphisms between finite Boolean lattices. There exists a model structure on our variant of cubical sets that is at once Quillen equivalent to and left induced from the classical model structure on simplicial sets along triangulation. This model structure is proper and hence its fibrations interpret Martin-Lof dependent types.

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arXiv2023

Cubical Approximation for Directed Topology II

The paper establishes an equivalence between directed homotopy categories of (diagrams of) cubical sets and (diagrams of) directed topological spaces. This equivalence both lifts and extends an equivalence between classical homotopy categories of cubical sets and topological spaces. Some simple applications include combinatorial descriptions and subsequent calculations of directed homotopy monoids and directed singular 1-cohomology monoids. Another application is a characterization of isomorphisms between small categories up to zig-zags of natural transformations as directed homotopy equivalences between directed classifying spaces. Cubical sets throughout the paper are taken to mean presheaves over the minimal symmetric monoidal variant of the cube category. Along the way, the paper characterizes morphisms in this variant as the interval-preserving lattice homomorphisms between finite Boolean lattices.

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arXiv2021

The Uniform Homotopy Category

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the respective Lipschitz and uniform settings. Cubical sets and uniform spaces admit the additional compatible structures of categories of (co)fibrant objects. A categorical equivalence between classical homotopy categories of cubical sets and spaces lifts to a full and faithful embedding from an associated Lipschitz homotopy category of cubical sets into an associated uniform homotopy category of uniform spaces. Bounded cubical cohomology generalizes to a representable theory on the Lipschitz homotopy category. Bounded singular cohomology on path-connected spaces generalizes to a representable theory on the uniform homotopy category. Along the way, this paper develops a cubical analogue of Kan's Ex^infinity functor and proves a cubical approximation theorem for uniform maps.

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arXiv2021

A Hurewicz Model Structure for Directed Topology

This paper constructs an h-model structure for diagrams of streams, locally preordered spaces. Along the way, the paper extends some classical characterizations of Hurewicz fibrations and closed Hurewicz cofibrations. The usual characterization of classical closed Hurewicz cofibrations as inclusions of neighborhood deformation retracts extends. A characterization of classical Hurewicz fibrations as algebras over a pointed Moore cocylinder endofunctor also extends. An immediate consequence is a long exact sequence for directed homotopy monoids, with applications to safety verifications for database protocols.

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arXiv2021

Invertibility in Category Representations

Inverse categories are categories in which every morphism x has a unique pseudo-inverse y in the sense that xyx=x and yxy=y. Persistence modules from topological data analysis and similarly decomposable category representations factor through inverse categories. This paper gives a numerical condition, decidable when the indexing category is finite, characterizing when a representation of a small category factors through an inverse category.

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arXiv2019

Triangulations of Conal Manifolds

Casual structure can take the form of cone bundles on a manifold, more general local preorders on a topological space, or simplicial orientations implicit in a simplicial set. This note takes a triangulation of a conal manifold M to mean an isomorphism between M and the locally preordered geometric realization of a simplicial set. Such triangulations completely encode causal and topological structure combinatorially. This note characterizes the triangulability of closed conal manifolds as the fibrewise freeness and generativity of the cone bundle.

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arXiv2016

Positive Alexander Duality for Pursuit and Evasion

Considered is a class of pursuit-evasion games, in which an evader tries to avoid detection. Such games can be formulated as the search for sections to the complement of a coverage region in a Euclidean space over a timeline. Prior results give homological criteria for evasion in the general case that are not necessary and sufficient. This paper provides a necessary and sufficient positive cohomological criterion for evasion in a general case. The principal tools are (1) a refinement of the Cech cohomology of a coverage region with a positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore homology of the coverage gaps with a positive cone encoding time orientation, and (3) a positive variant of Alexander Duality. Positive cohomology decomposes as the global sections of a sheaf of local positive cohomology over the time axis; we show how this decomposition makes positive cohomology computable as a linear program.

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arXiv2014

Flow-Cut Dualities for Sheaves on Graphs

This paper generalizes the Max-Flow Min-Cut (MFMC) theorem from the setting of numerical capacities to sheaves of partial semimodules over semirings on directed graphs. Motivating examples of partial semimodules include probability distributions, multicommodity capacity constraints, and logical propositions. Directed (co)homology theories for such sheaves describes familar constructs on networks. First homology classifies locally decomposable flows, an orientation sheaf over a semiring generalizes directions, connecting maps for homology assign values to flows, connecting maps for cohomology assign values to cuts, and a Poincare Duality describes a decomposition of flows as local flows over cuts. A consequent interpretation of feasible flow-values as a homotopy limit generalizes MFMC for edge weights in certain ordered monoids [Frieze] and hence also classical MFMC. Certain duality gaps are explained as a failure for directed sheaf (co)homology to satisfy a natural generalization of exactness.

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arXiv2011

Covering space theory for directed topology

The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a "local preorder" encoding control flow. In the case where time does not loop, the "locally preordered" state space splits into causally distinct components. The set of such components often gives a computable invariant of machine behavior. In the general case, no such meaningful partition could exist. However, as we show in this note, the locally preordered geometric realization of a precubical set admits a "locally monotone" covering from a state space in which time does not loop. Thus we hope to extend geometric techniques in static program analysis to looping processes.

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arXiv2010

Cubical approximation for directed topology I

Topological spaces - such as classifying spaces, configuration spaces and spacetimes - often admit extra temporal structure. Qualitative invariants on such directed spaces often are more informative yet more difficult to calculate than classical homotopy invariants on underlying spaces because directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Directed spaces often arise as geometric realizations of simplicial sets and cubical sets equipped with temporal structure encoding the orientations of simplices and 1-cubes. In an attempt to develop calculational tools for directed homotopy theory, we prove appropriate simplicial and cubical approximation theorems. We consequently show that geometric realization induces an equivalence between weak homotopy diagram categories of cubical sets and directed spaces and that its right adjoint satisfies an excision theorem. Along the way, we give criteria for two different homotopy relations on directed maps in the literature to coincide.

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arXiv2008

A convenient category of locally preordered spaces

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is Cartesian closed and that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.

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arXiv2008

Criteria for homotopic maps to be so along monotone homotopies

The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of abstract spacetime, we identify criteria for classically homotopic, monotone maps of pospaces to future homotope, or homotope along homotopies monotone in both coordinates, to a common map. We show that consequently, a hypercontinuous lattice equipped with its Lawson topology is future contractible, or contractible along a future homotopy, if its underlying space has connected CW type.

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