Since the first draft of this text, which appeared in 2012 on the arXiv, the theory of higher
Segal spaces has been further developed in various directions by several groups of authors …

We study the category Rep (Q, F 1) Rep(Q,F_1) of representations of a quiver Q over “the
field with one element”, denoted by F 1 F_1, and the Hall algebra of Rep (Q, F 1) …

This is an introduction to Hall algebras from the perspective of $2 $-Segal spaces or
decomposition spaces, as introduced by Dyckerhoff and Kapranov and G\'{a} lvez-Carrillo …

We outline a formal framework, based on functorial submaps (briefly fsm) and mappings
(called fsm-transfers) between them, to translate at a categorical level the study of maps …

We associate to a projective n-dimensional toric variety X∆ a pair of cocommutative (but
generally non-commutative) Hopf algebras Hα X, HT X. These arise as Hall algebras of …

Proto-exact categories, introduced by Dyckerhoff and Kapranov, are a generalization of
Quillen exact categories which provide a framework for defining algebraic K-theory and Hall …

Green's theorem states that the Hall algebra of the category of representations of a quiver
over a finite field is a twisted bialgebra. Considering instead categories of orthogonal or …

We introduce a general class of combinatorial objects, which we call multi-complexes, which
simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta …

We study the algebraic $ K $-theory and Grothendieck-Witt theory of proto-exact categories,
with a particular focus on classes of examples of $\mathbb {F} _1 $-linear nature. Our main …

This paper investigates infinite matroids from a categorical perspective. We prove that the
category of infinite matroids is a proto-exact category in the sense of Dyckerhoff and …