PostDoctoral Research Fellowship

Project Report

My work concerns an area of pure mathematics called category theory and its applications to homotopy theory, the study of mathematical objects up to a path of deformations. Over the duration of this award, I coauthored 10 papers and wrote one book, comprised of lecture notes for a graduate topics course I taught at Harvard (Categorical Homotopy Theory, Cambridge University Press 2014). Aside from the broader impacts of my research (in particular, a joint project with Dominic Verity that aims to make the theory of quasi-categories much easier to learn), I also initiated an online graduate reading course in category theory, meeting biweekly with twelve students from around the world over a six-month period. Category theory is a cross-disciplinary language for mathematics that describes general phenomena and is intended to serve as a simplifying force and useful abstraction. The principles of category theory, particularly the idea of characterizing an object by its universal property (meaning that transformations to or from the object correspond to particular mathematical structure), can suggest new mathematical definitions. In experience, such definitions are simpler than ad hoc constructions, yet powerful enough to prove desired theorems. In certain mathematical disciplines such as homotopy theory, many of the basic objects of study would be impossible to define without the use of category-theoretic notions. My research introduced new definitions that (1) described previously unobserved mathematical phenomena (e.g., parameterized mates and homotopy coherent adjunctions), (2) developed new tools with which to work in homotopy theory (e.g., derived resolutions), and (3) supplied new simplified proofs, re-grounding weak higher category theory ("brave new homotopy theory'') with foundations that are easier to learn and enabling the proof of new theorems. My work has led to a number of applications. The "algebraic'' perspective on homotopy theory, first introduced in my thesis, has been used to construct functorial factorizations in topological categories and in categories of modules over a differential graded algebra. Related ideas enable the construction of derived resolutions for Quillen adjunctions. My work has also inspired two papers written by current graduate students. An ongoing collaboration with Dominic Verity, which has produced four papers so far, aims to reconstruct the foundations of the theory of quasi-categories, an increasingly popular approach to abstract homotopy theory. They are higher-dimensional "(∞,1)-categories'' whose cells explicitly present the "higher homotopical information" invisible to homotopy categories, in which data is only recorded "up to homotopy." Ideas that were previously expressed as properties (e.g., "paths are composable up to homotopy") are reconceived as structures (a specific homotopy exhibiting a path as a composite, itself well-defined up to higher homotopy). Such compatibly-defined higher homotopical information is called "homotopy coherent." Any abstract "homotopy theory" is modeled by a quasi-category. Quasi-categories may be used to prove theorems in cases in which point-set models are intractable or unavailable. Our work offers a new approach to proofs of the foundational categorical results for quasi-category theory. Our methods use formal category theory to make it easier to understand these results and the means by which we prove them. This also leads to new theorems, in principle provable through traditional methods, but much more approachable via ours. I will describe a small portion of this project in more detail. It has a new theorem, a much-simplified proof of an existing theorem, and indicates directions for further work that is currently in progress. Adjoint functors are pervasive in mathematics, capturing an otherwise difficult to articulate duality, e.g., between "free'' and "forgetful'' constructions. Quillen adjunctions, particularly important to abstract homotopy theory because they capture the notion of morphism between model categories, give rise to adjunctions between the associated quasi-categories. A classical categorical theorem describes the "free adjunction," a 2-category ADJ with the property that 2-functors ADJ —> K correspond to adjunctions in the 2-category K. The 2-category ADJ can be regarded as a simplicial category, in this case an "(∞,2)-category,'' which we give the same name. Theorem. The simplicial category ADJ is the free homotopy coherent adjunction. Any adjunction of quasi-categories extends to a simplicial functor with domain ADJ, and moreover such extensions are homotopically unique. Thanks to a graphical calculus that we introduce, the data of a homotopy coherent adjunction is easy to describe: an n-dimensional arrow in ADJ is encoded by a strictly undulating squiggle on n+1 lines. We use our graphical calculus to prove the "freeness'' of the homotopy coherent adjunction ADJ. The surprise is that our model is so small; one would expect a (much larger) cofibrant replacement would be necessary to obtain the free homotopy coherent adjunction. Use of the free homotopy coherent adjunction enables a proof of the monadicity theorem in any (∞,2)-category. Work in progress aims to develop all of category theory at this level of generality.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Bruce P. Palka
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Riehl Emily
United States
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