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- Michael Shulman
- 2007

In some bicategories, the 1-cells are ‘morphisms’ between the 0cells, such as functors between categories, but in others they are ‘objects’ over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the ‘morphisms between 0-cells’, such as ring homomorphisms, are… (More)

In the United States, passenger vehicle manufacturers have been working together, along with the U. S. government, to study wireless communications for vehicle safety applications. From 2002-4, seven automotive manufacturers— BMW, DaimlerChrysler, Ford, GM, Nissan, Toyota, and VW— worked with the United States Department of Transportation (USDOT) to… (More)

- Michael Shulman
- Mathematical Structures in Computer Science
- 2015

We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by… (More)

- Michael Shulman
- 2006

Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equivalence. Our second goal is to generalize this result to enriched… (More)

- Daniel R. Licata, Michael Shulman
- 2013 28th Annual ACM/IEEE Symposium on Logic in…
- 2013

Recent work on homotopy type theory exploits an exciting new correspondence between Martin-Lof's dependent type theory and the mathematical disciplines of category theory and homotopy theory. The mathematics suggests new principles to add to type theory, while the type theory can be used in novel ways to do computer-checked proofs in a proof assistant. In… (More)

- Benedikt Ahrens, Krzysztof Kapulkin, Michael Shulman
- Mathematical Structures in Computer Science
- 2015

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of “category” for which equality and equivalence of categories agree. Such categories satisfy a version of the Univalence Axiom, saying that the type… (More)

- Michael Shulman
- 2008

We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category whose vertical and horizontal arrows are left and right Quillen functors, respectively, and that passage to derived… (More)

- Michael Shulman
- 2012

We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κ-ary exact categories” are a reflective sub-2-category of “κ-ary sites”, for any regular cardinal κ. A κ-ary exact category is an exact category with disjoint and universal κ-small coproducts, and a κ-ary site is a site whose covering sieves… (More)

- Egbert Rijke, Michael Shulman, Bas Spitters
- ArXiv
- 2017

Univalent homotopy type theory (HoTT) may be seen as a language for the category of ∞-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of factorization systems, reflective subuniverses, and modalities in homotopy type theory, including their construction… (More)

- VIGLEIK ANGELTVEIT, Michael Shulman, Justin Noel
- 2008

We define the notion of an enriched Reedy category, and show that ifA is a C-Reedy category for some symmetric monoidal model category C and M is a C-model category, the category of C-functors and C-natural transformations from A to M is again a model category.