Proto-exact categories of matroids, Hall algebras, and K-theory

Szczesny, Maciej; Jun, Jaiung; Eppolito, Chris

doi:10.1007/s00209-019-02429-z

Cited by 13 publications

(19 citation statements)

References 25 publications

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“…In this paper we define a category Mat • of infinite matroids which contains the category of finite matroids and strong maps as a full subcategory. Theorem 3.9 proves that Mat • has the structure of a proto-exact category, thereby generalizing [EJS20,Theorem 5.11]. Corollary 4.4 characterizes finitary matroids as the co-limits of finite matroids; along the way we obtain Corollary 4.6, yielding a "finitization" functor Mat • → Mat fin…”

Section: Introductionmentioning

confidence: 62%

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On Infinite Matroids with Strong Maps: Proto-exactness and Finiteness Conditions

Eppolito,

Jun

2021

Preprint

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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 Kapranov, thereby generalizing our previous result on the category of finite matroids. We also characterize finitary matroids as co-limits of finite matroids, and show that the finite matroids are precisely the finitely presentable objects in this category.

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Section: Introductionmentioning

confidence: 62%

“…See [EJS20] for examples and motivation regarding proto-exact categories. For proto-exact categories in connection with algebraic geometry, see [ELY20; JS20].…”

Section: Proto-exact Categoriesmentioning

confidence: 99%

On Infinite Matroids with Strong Maps: Proto-exactness and Finiteness Conditions

Eppolito,

Jun

2021

Preprint

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“…Roughly speaking, a proto-exact category is a pointed category with two distinguished classes of morphisms (admissible monomorphisms and admissible epimorphisms) satisfying certain conditions on pullback and pushout diagrams from which one can obtain a notion of admissible exact sequences. 1 Several interesting "combinatorial" categories are equipped with a proto-exact structure, for instance, the category of matroids [EJS20], the category of representations over a quiver (and more generally any monoid) over "the field with one element" [Szc12], [JS20a], [JS21]. Categories with more algebro-geometric flavors, which are not additive, have been explored in [Szc18], [JS20b], [ELY20].…”

Section: Introductionmentioning

confidence: 99%

“…2 (2) One can define a well-behaved version of algebraic K-theory for A (either through Quillen's Q-construction or Waldhausen's S-construction -see [DK12,ELY20,Hek17]. Even for relatively simple combinatorial categories A this is a rich and interesting invariant [CLS12,EJS20].…”

Section: Introductionmentioning

confidence: 99%

Proto-exact categories of modules over semirings and hyperrings

Jun¹,

Szczesny²,

Tolliver³

2022

Preprint

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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 algebras in a non-additive setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor.In this paper, we show that the categories of modules over semirings and hyperrings -algebraic structures which have gained prominence in tropical geometry -carry proto-exact structures.In the first part, we prove that the category of modules over a semiring is equipped with a protoexact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices L has a proto-exact structure, and furthermore that the subcategory of L consisting of finite lattices is equivalent to the category of finite B-modules as protoexact categories, where B is the Boolean semifield. We also discuss some relations between L and geometric lattices (simple matroids) from this perspective.In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the Krasner hyperfield K, a well-known relation between finite K-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.

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“…The first realization is that of representations of a quiver over the so-called field with one element F 1 . The Hall algebra H Q,F 1 was first studied by Szczesny [24], who, together with collaborators, has also studied a number of other combinatorial-type Hall algebras [13], [23], [25], [6], [26]. See also [8].…”

Section: Introductionmentioning

confidence: 99%

Degenerate versions of Green's theorem for Hall modules

Young¹

2018

Preprint

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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 symplectic quiver representations leads to a class of modules over the Hall algebra, called Hall modules, which are also comodules. A module theoretic analogue of Green's theorem, describing the compatibility of the module and comodule structures, is not known. In this paper we prove module theoretic analogues of Green's theorem in the degenerate settings of finitary Hall modules of Rep F1 (Q) and constructible Hall modules of Rep C (Q). The result is that the module and comodule structures satisfy a compatibility condition reminiscent of that of a Yetter-Drinfeld module.

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Proto-exact categories of matroids, Hall algebras, and K-theory

Cited by 13 publications

References 25 publications

On Infinite Matroids with Strong Maps: Proto-exactness and Finiteness Conditions

On Infinite Matroids with Strong Maps: Proto-exactness and Finiteness Conditions

Proto-exact categories of modules over semirings and hyperrings

Degenerate versions of Green's theorem for Hall modules

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