Research

I am interested in Algebraic Geometry, Homotopy Theory, and Category Theory, in particular how the latter informs the former two.

In more details, I am interested in higher algebra, which is, in a broad sense, the study of familiar algebraic structures—groups, rings, and others—suitably translated to (higher) categories. These gadgets concretely manifest themselves, for example, in Intersection Theory and $K$-Theory, and the geometry of homotopy types.

I am also intersted in computer assisted formalization of Mathematics, mostly in the context of Homotopy Type Theory.

Projects

• Categorification of algebraic cycles and Intersection Theory (With Niranjan Ramachandran and Maxime Ramzi).
  We aim to assign to an algebraic cycle a geometric object in a way that generalizes the well known correspondence for divisors (i.e. codimension one cycles).

  If $X$ is a smooth proper variety over a field $F$, and $\alpha \in Z^p(X)$ is a cycle of codimension $p$, we assign to it a higher stack $\mathcal{C}_\alpha$ by way of a suitable classifying map $$f_\alpha \colon X \longrightarrow \mathrm{B}^p\mathcal{K}_{p,X}\,,$$ where $\mathcal{K}_{p,X}$ is the $p$-th $K$-Theory sheaf on $X$. So $\mathcal{C}_\alpha$ is a $\mathrm{B}^{p-1}\mathcal{K}_{p,X}$-torsor that we construct as a sheaf of animas on $X$. We aim to prove, for example, that this construction is compatible with intersection, that is, the torsor $\mathcal{C}_{\alpha\cdot \beta}$ associated to the intersection $\alpha\cdot \beta$ of two cycles is equivalent to the smash $\mathcal{C}_\alpha \wedge \mathcal{C}_\beta$ coming from the cup product and the smash product in $K$-Theory.

  The construction of $\mathcal{C}_\alpha$, or, equivalently, the classifying map $f_\alpha$, relies on the so-called Gersten conjecture, which is true for our $X$, asserting that $\mathcal{K}_{p,X}$ has a certain explicit resolution, offering a convenient model for $\mathrm{B}^p\mathcal{K}_{p,X}$. An important goal is to achieve an independent construction of $f_\alpha$.

  This is based on the case $p=2$ we previously established (see here and here), where the torsor $\mathcal{C}_\alpha$ is a $\mathcal{K}_{2,X}$-gerbe.

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• Cubical constructions and Goodwillie Calculus for exact ∞-categories (With Arash Karimi).
  A cubical complex devised by Eilenberg and MacLane to compute the stable homology of abelian group, the so-called $Q$-construction, was extended to exact categories by R. McCarthy. If $F\colon \mathbf{Ex}\to \mathcal{A}$ is a functor from the category of exact categories to a fixed abelian category, then, for any exact category $\mathcal{E}$ one has a complex $Q_F(\mathcal{E})$ whose homology is related to the Goodwillie derivative of $F$; depending on the choice of $F$, it computes interesting homologies related to the (Waldhausen) $K$-theory spectrum of $\mathcal{E}$.

  Our work generalizes this circle of ideas to the ∞-categorical case, where now $F\colon \mathbf{Ex}_∞\to \mathcal{C}$ is is a fixed functor from the ∞-category of exact ∞-categories, and $\mathcal{C}$ is a fixed stable ∞-category. The output of our construction, for any exact ∞-category $\mathcal{E}$, is a filtered object $Q_F(\mathcal{E})$ of $\mathcal{C}$. As a result, there is a genuine complex in the homotopy category $h\mathcal{C}$, and more interesting things happen when $\mathcal{C}$ is equipped with a $t$-structure.

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• Multiplicative comparison of $K$-Theories, and homological functors (With Brandon Doherty, Milind Gunjal, Arash Karimi).
  Bohmann and Osorno established a multiplicative equivalence between multiplicative Walhausen’s and Segal’s $K$-theories at the level of multicategories. Specifically, they established a full multi-natural transformation between multi-functors $$\mathbf{Wald} \longrightarrow \mathbf{SMC} \overset{\Gamma(\cdot)}{\longrightarrow} \mathrm{Sp}(s\mathbf{Cat}_*)$$ and $$\mathbf{Wald} \overset{S_\bullet}{\longrightarrow} \mathrm{Sp}(s\mathbf{Cat}_*)$$ The first is Elmendorff and Mandell’s version of Segal’s construction (I use $\Gamma$ for that, for Gamma-space), and the second is Blumberg and Mandell’s version of Waldhausen’s $S$-construction. $\mathbf{Wald}$ and $\mathbf{SMC}$ are the (multi)categories of Waldhausen and symmetric monoidal categories, respectively, and $\mathrm{Sp}(s\mathbf{Cat}_*)$ that of simplicial pointed categories. We add a fourth vertex, $\mathrm{Sp} (\mathbf{Ch}_{≥0}(\mathcal{A}))$, the category of spectra in connective chain complexes in an abelian category $\mathcal{A}$, together with a fixed functor $F\colon \mathbf{Cat}_*\to \mathcal{A}$, providing homology theories for $\mathbf{Wald}$, $\mathbf{SMS}$, and $\mathrm{Sp}(s\mathbf{Cat}_*)$. The goal is to prove that we obtain a commutative tetrahedron where the remaining edges $\mathbf{Wald} \to \mathrm{Sp} (\mathbf{Ch}_{≥0}(\mathcal{A}))$ and $\mathbf{SMC}\to \mathrm{Sp} (\mathbf{Ch}_{≥0}(\mathcal{A}))$ are given by MacLane’s cubical $Q$-constructions previously used by McCarthy and Burgos Gil-Wang (for Waldhausen categories) and Cindy Lester and myself (for symmetric monoidal categories).

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• Determinant functors for Waldhausen and Triangulated categories (With Yaineli Valdes and Cindy Lester).

  Determinant functors, following the pioneering work by Deligne, have been defined for Waldhausen as well as triangulated categories. Computing models of these is particularly relevant, as it allows to compute universal determinant functors, and hence 1-types of the relevant $K$-theory spectra.

  A particularly interesting question is the determinat functor’s behavior relative to biexact functors of the input categories; for example, if $F\colon \mathcal{C}\times \mathcal{D}\to \mathcal{E}$ is a biexact functor of Waldhausen categories, or if $\otimes \colon \mathcal{T}\times \mathcal{T}\to \mathcal{T}$ is the tensor operation in a triangulated category.

  We adopt a multi-categorical point of view to answer these questions, based on Y. Valdes’ thesis and work by Cindy Lester and myself, where cubical complexes again play a part.

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• The 2-Type of the $K$-Theory spectrum of a Waldhausen Category (With Milind Gunjal).

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