Thomason's colimit theorem for the double category of elements (with Andrew Gill)
The category of elements plays a central role in category theory, connecting presheaves and discrete fibrations, as well as providing a characterization of representable functors in terms of terminal objects in the category of elements. This construction also plays an important role in homotopy theory, mainly due to Thomason’s colimit theorem, expressing the fact that the Grothendieck construction of a diagram of categories gives a categorical model for the homotopy type of the homotopy colimit of the diagram of categories.
Various authors have worked to this construction to the 2-categorical setting, taking a 2-functor and producing a 2-category of elements. Interestingly, this 2-categorical version also satisfies Thomason’s colimit theorem, thus showing a good homotopical behavior. However, its categorical behavior is somewhat mixed. The solution is to instead consider a double category of elements, introduced by Grandis and Paré, which now has all the desired categorical properties. In this article, we study the double category of elements from a homotopical perspective, showing that it satisfies Thomason’s colimit theorem, and hence it also models the correct homotopy type.
A combinatorial construction of homology via ACGW categories (with Brandon Shapiro and Inna Zakharevich)
This is an expository paper written for the proceedings of the workshop “Higher Segal Spaces and their Applications to Algebraic K-Theory, Hall Algebras, and Combinatorics”. Here, we attempt to introduce the reader to the key ideas and techniques for working with (A)CGW categories. To do so, we focus on how homology theory generalizes to ACGW categories, particularly in the example of finite sets. We show how the ACGW formalism can be used to produce various classical homological algebra results such as the Snake lemma and long exact sequences of relative pairs.
We define an S•-construction for squares categories, and study the conditions required for it to model the correct K-theory space. When the squares categories are "stable", we show that this S•-construction produces a 2-Segal space.
An appendix by Maxine Calle (based on her joint work with Liam Keenan) discusses the challenges that arise when trying to adopt Waldhausen's formulation of the Additivity Theorem to the squares setting.
It is well-known that the stable model structure on symmetric spectra cannot be transferred from the one on sequential spectra through the forgetful functor. We use the fibrant transfer theorem of Guetta--Moser--Sarazola--Verdugo to show it can be transferred between fibrant objects, providing a new, short and conceptual proof of its existence. In particular, this allows us to only use weak (stable) homotopy equivalences instead of the more complicated class of stable weak equivalences of symmetric spectra, since these agree between fibrant objects!
We introduce a version of the Grothendieck construction that takes V-enriched functors as inputs, and produces categories internal to V as outputs. This generalizes the classical category of elements construction by taking V=Set. As in the classical version, we prove that terminal objects in this Grothendieck construction can be used to characterize V-representable functors; that it gives an equivalence between V-presheaves and the appropriate notion of internal discrete fibrations; and use it to show that enriched weighted limits can always be computed as conical limits in the internal setting.
Equivariant trees and partition complexes (with Julie Bergner, Peter Bonventre, Maxine Calle, and David Chan)
We introduce two definitions of G-equivariant partitions of a finite G-set, both of which yield G-equivariant partition complexes. By considering suitable notions of equivariant trees, we show that G-equivariant partitions and G-trees are G-homotopy equivalent, generalizing existing results for the non-equivariant setting. Along the way, we develop equivariant versions of Quillen's Theorems A and B, which are of independent interest.
We develop new techniques for constructing model structures from a given class of fibrant objects and a choice of weak equivalences between them. As a special case, we obtain a more flexible version of the classical right-induction theorem in the presence of an adjunction. Namely, instead of lifting the classes of fibrations and weak equivalences through the right adjoint, we now only do so between fibrant objects, which allows for a wider class of applications.
We use these new tools to define a new model structure that makes the square functor Sq:2Cat→DblCat into a Quillen equivalence, providing a double categorical model for the homotopy theory of 2-categories.
We define new double categorical structures that modify the CGW-categories of Campbell and Zakharevich, to obtain a framework called ECGW categories that allows for an S•-construction akin to Waldhausen's, and show how it produces a K-theory spectrum which satisfies an analogue of the Additivity Theorem. We also define a notion of "relative ECGW categories" which have weak equivalences determined by a subcategory of acyclic objects satisfying minimal conditions; these satisfy analogues of the Fibration and Localization Theorems that generalize previous versions in the literature. We illustrate these results with examples including exact categories, extensive categories, algebraic varieties, and polytopes up to scissors congruence.
A model structure for Grothendieck fibrations (with Lyne Moser)
Journal of Pure and Applied Algebra (2024), arxiv
We construct two model structures, whose fibrant objects capture the notions of discrete fibrations and of Grothendieck fibrations over a category C. For the discrete case, we build a model structure on the slice Cat/C, Quillen equivalent to the projective model structure on C-presheaves via the classical category of elements construction. A marked version of the Grothendieck construction gives the appropriately analogous result for the cartesian case. Finally, we show that both of these model structures have the expected interactions with their ∞-counterparts.
Cofibration category of digraphs for path homology (with Daniel Carranza, Brandon Doherty, Chris Kapulkin, Morgan Opie, and Liang Ze Wong)
Algebraic Combinatorics (2024), arxiv
We prove that the category of directed graphs and graph morphisms carries a cofibration category structure in which the weak equivalences are the graph morphisms inducing isomorphisms on path homology.
A model structure for weakly horizontally invariant double categories (with Lyne Moser and Paula Verdugo)
Algebraic and Geometric Topology (2023), arxiv
We construct a model structure on DblCat which is monoidal with respect to the Gray tensor product, and contains the homotopy theory of 2-categories. The fibrant objects are the weakly horizontally invariant double categories, and the weak equivalences between fibrant objects are given by the "double biequivalences", a double-categorical version of biequivalences.
A 2Cat-inspired model structure for double categories (with Lyne Moser and Paula Verdugo)
Cahiers de Topologie et Géométrie Différentielle Catégoriques (2022) , arxiv
We construct a model structure on the category DblCat of double categories. Unlike previous model structures, it recovers the homotopy theory of 2-categories through the horizontal embedding H: 2Cat → DblCat, which is now a left and right Quillen functor, and homotopically fully faithful.
Want to see what it's about? Here are slides and a recording of my talk at the MIT Categories Seminar in 2020, the first half of which is devoted to this paper.
Stable homotopy hypothesis in the Tamsamani model (with Lyne Moser, Viktoriya Ozornova, Simona Paoli, and Paula Verdugo)
Topology and its Applications (2022), arxiv
The homotopy hypothesis is a well-known bridge between category theory and algebraic topology. It basically states that n-types (that is, spaces whose i-th homotopy groups are trivial for i greater than n) can be modeled by weak n-groupoids. In turn, the stable homotopy hypothesis proposes a similar result, only it focuses on spectra instead of spaces. In this paper, we prove that symmetric monoidal weak n-groupoids in the Tamsamani model provide a model for stable n-types.
Want to see what it's about? Here are slides and a recording of my talk on this paper at UNAM's Category Theory Seminar in 2021.
Cotorsion pairs and a K-theory localization theorem
Journal of Pure and Applied Algebra (2020); arxiv
Inspired by Hovey's result relating cotorsion pairs to model category structures, we study the relation between cotorsion pairs and Waldhausen category structures. This allows us to prove a new version of Quillen’s Localization Theorem, relating the K-theory of exact categories A ⊆ B to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed trough a cotorsion pair.
Want to see what it's about? Here are slides and a recording of my talk on this paper at the Purdue Topology Seminar in 2021.
A recipe for black box functors (with Brendan Fong)
Theory and Applications of Categories (2020), arxiv
Reviewing a principled method for constructing hypergraph categories and functors, known as decorated corelations, in this paper we construct a category of decorating data, and show that the decorated corelations method is itself functorial, with a universal property characterised by a left Kan extension. We then show that every hypergraph category is equivalent to a decorated corelations category, and thus argue that the category of decorating data is a good setting in which to construct any hypergraph functor, giving a new construction of Baez and Pollard's black box functor for reaction networks as an example.
Want to see what it's about? Here are slides for my talks on this paper at the CT and ACT conferences in 2019.
My phd thesis! Go check it out.
Dissertation for my Bachelor in Mathematics (done in Uruguay); abstract in English.