9 papers
Discrete homotopy hypothesis for n-types
Daniel Carranza, Chris Kapulkin
We show that discrete and classical homotopy theories are equivalent after localizing at n-equivalences for any non-negative integer n. By constructing an explicit homotopy inverse…
Diagonal Lemma for Presheaves on Eilenberg-Zilber Categories
Daniel Carranza, Chris Kapulkin, Liang Ze Wong
The diagonal lemma asserts that if a map of bisimplicial sets is a levelwise weak equivalence in the Kan-Quillen model structure, then it induces a weak equivalence of the diagonal…
Cofibration category of digraphs for path homology
Daniel Carranza, Brandon Doherty, Chris Kapulkin +3
We prove that the category of directed graphs and graph maps carries a cofibration category structure in which the weak equivalences are the graph maps inducing isomorphisms on pat…
Cubical setting for discrete homotopy theory, revisited
Daniel Carranza, Chris Kapulkin
We construct a functor associating a cubical set to a (simple) graph. We show that cubical sets arising in this way are Kan complexes, and that the A-groups of a graph coincide wit…
Homotopy groups of cubical sets
Daniel Carranza, Chris Kapulkin
We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that the…
Cubical models of -presheaves and the Bousfield-Kan formula
Kensuke Arakawa, Daniel Carranza, Chris Kapulkin
We construct the covariant and the cocartesian model structures on the slice categories of cubical sets and marked cubical sets, respectively. As an application, we derive a versio…