Recent years have seen several attempts to formulate notions of algebraic geometry in characteristic one based on monoids, semirings, hyperrings, and blueprints , , , , . One reason for this effort is the desire to develop scheme-theoretic foundations for tropical geometry , . In these “exotic” theories, just as in “ordinary” algebraic geometry, an affine scheme corresponds one of the aforementioned algebraic structures, which we generically denote A for the moment. Proceeding by analogy, one expects a quasi-coherent sheaf on to correspond to some appropriate notion of A-module, and is therefore led to study the category A-mod of A-modules and its homological properties in general. Here, the situation is complicated by the fact that unlike the case of commutative rings, A-mod is typically not abelian or even additive, and so what should be considered an “exact sequence” does not have an immediate answer.
Proto-exact categories, introduced by T. Dyckerhoff and M. Kapranov [DK12] as a generalization of Quillen exact cateogires, provide a flexible framework for exact sequences in (potentially) non-additive categories. 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].
There are at least two “features” associated with a proto-exact category :
If is finitary, in the sense that the number of inequivalent short exact sequences is finite for each pair of objects , then one may define the Hall algebra . This is an associative (and in “good” cases Hopf) algebra spanned by the isomorphism classes of objects of , whose structure coefficients count the number of extensions between objects. Classically - for instance when is the category of quiver representations or coherent sheaves on a curve over over a finite field , , yields quantum groups and related objects and is an important tool in their representations theory. When is non-additive, typically has a combinatorial flavor. In this case, the Hall algebra often becomes a Hopf algebra, where the product of two objects is obtained by “assembling” two objects into a new object and coproduct encodes all possible ways to “disassemble” the given object into two objects. As an application, one may study various operations and identities for combinatorial objects from the Hall algebra perspective.2
One can define a well-behaved version of algebraic K-theory for (either through Quillen's Q-construction or Waldhausen's S-construction - see , , . Even for relatively simple combinatorial categories this is a rich and interesting invariant , .
The main goal of the current paper is to enlarge the catalogue of non-additive proto-exact categories by showing these include the categories of modules over semirings as well as hyperrings. Modules over an idempotent semiring are closely related to matroid theory [GG18] and modules over a hyperring have an interesting connection to finite incidence geometries [CC10a], [Jun18b] and matroids [BB19]. We also examine the category of algebraic lattices in relation to finite modules over , and discuss how the proto-exact structure of algebraic lattices is related to the proto-exact structure of the category of matroids in [EJS20] via geometric lattices. Along the way, we also investigate whether the proto-exact categories constructed are finitary, and thus whether the Hall algebra is defined.
Let R be a semiring and M be an R-module. We define admissible monomorphisms (resp. admissible epimorphisms) to be equalizers (resp. coequalizers) of some morphisms and the zero map. We first prove the following.
Theorem A Theorem 3.14
Theorem A Theorem 3.14
Let R be a semiring. With the admissible monomorphisms and admissible epimorphisms as above, the category of modules over R is a proto-exact category.
Next, we turn our attention to the category of algebraic lattices (not necessarily finite). Recall that an algebraic lattice is a complete lattice such that any element is a join of compact elements. In particular, any finite lattice is an algebraic lattice. For algebraic lattices, roughly we define admissible monomorphisms (resp. admissible epimorphisms) to be downward closed subsets (resp. upward closed subsets). See Definition 4.5 for the precise definition. We prove the following.
Theorem B Theorem 4.15 With the admissible monomorphisms and admissible epimorphisms as above, the category of algebraic lattices is a proto-exact category.
Theorem B Theorem 4.15
With the admissible monomorphisms and admissible epimorphisms as above, the category of algebraic lattices is a proto-exact category.
Let be the subcategory of consisting of finite lattices. is a proto-exact category with the induced proto-exact structure (from . Also, from Theorem A, one can prove that the category of finite -modules is proto-exact with the induced proto-exact structure (from ). We prove the following.
Theorem C Corollary 4.22 The category is equivalent to as proto-exact categories.
Theorem C Corollary 4.22
The category is equivalent to as proto-exact categories.
A geometric lattice is a finite semimodular lattice in which every element is a join of atoms. Geometric lattices provide another cryptomorphic definition for (simple) matroids. In fact, the subcategory of consisting of geometric lattices is a proto-exact subcategory of (Proposition 4.27). Note that the category contains “more morphisms” than the category of matroids with strong maps as in [EJS20]. In particular, the proto-exact structure on is different from the proto-exact structure for the category of matroids studied in [EJS20]. For more details, see Section 4.3.
Next, we move to hyperrings. A morphism of hyperrings preserves multi-valued addition in a “weak sense” (see Definition 2.10). For modules over hyperrings, we define admissible monomorphisms (resp. admissible epimorphisms) to be injective (resp. surjective) morphisms which preserve multi-valued addition in a “strong sense”. Then, we prove the following.
Theorem D Theorem 5.11 Let H be a hyperring. With the admissible monomorphisms and admissible epimorphisms as above, the category of modules over H is a proto-exact category.
Theorem D Theorem 5.11
Let H be a hyperring. With the admissible monomorphisms and admissible epimorphisms as above, the category of modules over H is a proto-exact category.
Acknowledgments The authors would like to thank Chris Eppolito for his helpful comments on the first draft of the paper.
In [Rin90], Ringel defined the Hall algebra of the category of representations of a quiver Q over a finite field , and proved that in the case of a simply laced Dynkin quiver Q, the associated Hall algebra is the upper triangular part of the quantum group which is classified by the same Dynkin diagram. Later in [Kap97], Kapranov investigated , the category of coherent sheaves on a smooth projective curve X over a finite field . and proved when X is a projective line, some
The category of modules over a semiring as a proto-exact category
In this section, we prove that the category of modules over a semiring is a proto-exact category. We define admissible monomorphisms (resp. admissible epimorphisms) to be equalizers (resp. coequalizers) in certain forms. Then, we give an equivalent description of admissible monomorphisms and admissible epimorphisms in terms of saturated submodules.
Let M be a module over a semiring R. Recall that a submodule N of M is said to be saturated if and implies that for any . It is
Algebraic lattices, -modules, and matroids
In this section, we focus on the case when , the Boolean semifield. It is well-known that any finite -module is naturally equipped with a lattice structure.6 See, for instance, [Les12, §2].
We prove that the category of algebraic lattices is proto-exact, and explore certain connections to -modules and the subcategory of geometric lattices (i.e. simple matroids).
The category of modules over a hyperring as a proto-exact category
In this section, we study the category of modules over a hyperring from the perspective of proto-exact categories. Throughout this section, let H be a hyperring and be the category of H-modules. We note that in [Mad06], certain categorical aspects of modules over a hyperring are studied, however it does not show that the category of modules over a hyperring is proto-exact.
Definition 5.1 Let be the class of strict injective homomorphisms in and be the class of strict surjective homomorphisms in
Let be the class of strict injective homomorphisms in and be the class of strict surjective homomorphisms in
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