Research

My PhD project approaches rational homotopy theory in the framework of model categories and simplicial sets, and it establishes connections of this approach with the study of categories up to homotopy, that is, (∞, 1)-categories.

My PhD thesis is on models for rational (∞, 1)-categories. I introduce rational (∞, 1)-categories, which are (∞, 1)-categories enriched in spaces whose higher homotopy groups are rational vector spaces, and I provide equivalent models for rational (∞, 1)-categories. Two such models are

1. rational complete Segal spaces, which are based on Rezk's complete Segal space model for (∞, 1)-categories; and

2. rational Segal categories, which are based on Bergner's models for (∞, 1)-categories using Segal categories.

Although my models hinge on my work on the rational homotopy theory of non-simply connected spaces using the toolkit of model categories, my methods work for (∞, 1)-categories enriched in general localizations of spaces.

The future directions of my work include the development of rational homotopy-theoretic and localized analogs of

1. other models for (∞, 1)-categories, such as

   1. Bergner's model using simplicial categories and

   2. Joyal and Lurie's model using quasi-categories;

2. models for (∞, n)-categories in the case n>1; and

3. models for (∞, 1)-operads.

I am also interested in producing an algebraic model for rational (∞, 1)-categories akin to the algebraic models for the rational homotopy theory of spaces.

In a recent paper, Heuts and Moerdijk develop a poset of trees whose classifying space is homotopy equivalent to the n-th partition complex, that is, the classifying space of the poset of partitions of a set with n elements. The latter poset features in work of Arone, Dwyer, and Lesh, and it finds a unitary analog introduced by Arone and Lesh. In joint work with Julie Bergner, Pedro Brunialti Lima de Andrade, and Josh Turner, we develop a topological poset of trees whose classifying space is homotopy equivalent to the n-th unitary partition complex, that is, the classifying space of the topological poset of partitions of n-dimensional complex space in pairwise orthogonal subspaces. Our work builds on the study of unitary partition complexes by Bergner, Joachimi, Lesh, Stojanoska, and Wickelgren, and it also entails some new results in the theory of topological categories that may be of independent interest.

Given a finite graph G, Dani and Mainkar associate with it a two-step nilpotent Lie algebra L(G), as well as a compact nilmanifold N(G). By a theorem of Nomizu, the de Rham cohomology of N(G) is isomorphic to the Lie algebra cohomology of L(G). In joint work with Marco Aldi, Samuel Bevins, Sergio Da Silva, Quincy Frias, and Tomás Mejía Gómez, we use Lie-algebraic methods to calculate the Betti numbers of all Dani-Mainkar nilmanifolds associated with path graphs. To assist with our computations, we develop a graphical calculus for the Chevalley-Eilenberg complex of the Lie algebra associated with any path graph.

Given a lattice L, Balchin, MacBrough, and Ormsby endow the set Tr(L) of transfer systems on L with a partial order ≤ such that intervals in the poset (Tr(L), ≤) correspond to composition-closed premodel structures on L. In ongoing joint work with Sanjana Agarwal and Ben Spitz, we show that, in the case when L is a finite lattice, the poset (Tr(L), ≤) is a lattice if and only if L is a fusion of finite total orders.

We introduce rational (∞, 1)-categories, which are (∞, 1)-categories enriched in spaces whose higher homotopy groups are rational vector spaces. We provide two models for rational (∞, 1)-categories, rational complete Segal spaces and rational Segal categories, and we show that they are equivalent.

In Quillen's paper on rational homotopy theory, the category of 1-reduced simplicial sets is endowed with a family of model structures, the most prominent of which is the one in which the weak equivalences are the rational homotopy equivalences and the fibrant objects are the rational Kan complexes. In this paper, we give a modern approach to this family of model structures. We recover Quillen's family of model structures by first left-transferring the model structure on pointed simplicial sets and then left Bousfield localizing at the rationalization maps of spheres. Applying this localization to the model category of all spaces yields a model category in which the weak equivalences are the rational homotopy equivalences in the extended sense of Gómez-Tato, Halperin, and Tanré and the fibrant objects are the rational spaces. Thus, we generalize Quillen's family of model structures beyond the rational homotopy theory of 1-connected spaces.

In this thesis, we introduce rational (∞, 1)-categories, which are (∞, 1)-categories enriched in spaces whose higher homotopy groups are rational vector spaces; such spaces are called rational. We provide two models for rational (∞, 1)-categories, rational complete Segal spaces and rational Segal categories, and we show that they are equivalent. Our methods work for (∞, 1)-categories enriched in general localizations of spaces, and we develop our arguments at that level of generality while occasionally touching base with our rational homotopy-theoretic case of special interest.

To develop our models for rational (∞, 1)-categories, we first produce a model category whose fibrant objects are the rational spaces. To that end, we give a modern perspective on Quillen's paper on rational homotopy theory, where Quillen provides a model category whose fibrant objects are the simply connected rational spaces. Specifically, we recover Quillen's model category as a left Bousfield localization, and we then apply the same localization to all spaces to get our desired generalization of Quillen's model category to non-simply connected spaces. Besides its usefulness for the purpose of modeling rational (∞, 1)-categories, our model category for rational spaces may be of independent interest in rational homotopy theory, for it encodes the rational homotopy theory of non-simply connected spaces developed by Gómez-Tato, Halperin, and Tanré.