Research

PAPERS/PREPRINTS:


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.



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.


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 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. 


We define new double categorical structures that modify the CGW-categories of Campbell and Zakharevich to include the data of weak equivalences. These structures are used to define chain complexes and quasi-isomorphisms for any category with suitably nice coproducts. In particular, we show how chain complexes of finite sets satisfy an analogue of the Gillet--Waldhausen Theorem: their K-theory agrees with the classical K-theory of finite sets, and thus provides a new model for the sphere spectrum. 


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. 

 

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.

 

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.

 

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.

 

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.


THESIS:

My phd thesis! Go check it out.

 

Dissertation for my Bachelor in Mathematics (done in Uruguay); abstract in English.