Emily Riehl

A Sharp Bound for the Degree of Proper Monomial Mappings Between Balls

The authors prove that a proper monomial holomorphic mapping from the two-ball to the N-ball has degree at most 2N-3, and that this result is sharp. The authors first show that certain group-invariant polynomials (related to Lucas polynomials) achieve the bound. To establish the bound the authors introduce a graph-theoretic approach that requires determining the number of sinks in a directed graph associated with the quotient polynomial. The proof also relies on a result of the first author that expresses all proper polynomial holomorphic mappings between balls in terms of tensor products.

Homotopical resolutions associated to deformable adjunctions

Given an adjunction FaG connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the completion and cocompletion associated to the monad and comonad of the adjunction. We discuss applications of these resolutions to spectral sequences for derived completions and Goodwillie calculus in general model categories. 55U35; 18G55, 18G10

On the construction of functorial factorizations for model categories

We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use “algebraic” characterizations of fibrations to produce factorizations that have the desired lifting properties in a completely categorical fashion. We illustrate these methods in the case of categories enriched, tensored and cotensored in spaces, proving the existence of Hurewicz-type model structures, thereby correcting an error in earlier attempts by others. Examples include the categories of (based) spaces, (based) G ‐spaces and diagram spectra among others. 55U35, 55U40; 18A32, 18G55

Kan extensions and the calculus of modules for ∞–categories

Various models of

On the structure of simplicial categories associated to quasi-categories

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On the intersections of polynomials and the Cayley-Bacharach theorem

-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an

Levels in the toposes of simplicial sets and cubical sets

\infty

Complicial Sets, an Overture

-cosmos. In a generic

Women’s History Month

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Made-to-Order Weak Factorization Systems

-cosmos, whose objects we call