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corey-state-dissertation-ohiolink.pdf (548.78 KB)

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Structure diagrams for symmetric monoidal 3-categories: a computadic approach

Staten, Corey
http://rave.ohiolink.edu/etdc/view?acc_num=osu1525455392722049

Abstract Details

2018, Doctor of Philosophy, Ohio State University, Mathematics.
The stable homotopy hypothesis in dimension 2 states that the homotopy theory of Picard 2-categories is equivalent to the homotopy theory of stable 2-types. Since Picard 2-categories are fully algebraic, they give us an algebraic context in which to study stable 2-types. The corresponding Picard 2-category versions of many standard constructions on stable 2-types are not yet well understood. In chapter 1, we examine symmetric monoidal bihomomorphisms C -> Sigma^2 B at the level of representing functors, where C is a Picard 1-category and B is an abelian group. In the case that C is an abelian group, our results provide an alternative proof of the known classification result for first k-invariants of Picard n-categories. We then turn our sights on symmetric monoidal trihomomorphisms between strict symmetric monoidal 3-categories, which would be the maps representing second k-invariants. A concrete data and axioms definition of such maps does not appear in the literature. In chapter 2, we provide a computadic framework for exploring potential structure cell boundaries and axioms for this and similar definitions. Along the way, we prove decomposition results concerning representations of cells in certain types of 3-computads, which are independently of interest. We provide an implementation of these results in an open source Python library, CatComputad, which is discussed in chapter 3. We then use this library to propose an extension of a structure cell family which should comprise part of the definition of the desired maps.
Niles Johnson (Advisor)
Jim Fowler (Committee Member)
Sanjeevi Krishnan (Committee Member)
94 p.
Mathematics
symmetric monoidal category; tricategory; trihomomorphism; computad; stable homotopy hypothesis

Recommended Citations

Citations

  • Staten, C. (2018). Structure diagrams for symmetric monoidal 3-categories: a computadic approach [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1525455392722049

    APA Style (7th edition)

  • Staten, Corey. Structure diagrams for symmetric monoidal 3-categories: a computadic approach. 2018. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1525455392722049.

    MLA Style (8th edition)

  • Staten, Corey. "Structure diagrams for symmetric monoidal 3-categories: a computadic approach." Doctoral dissertation, Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1525455392722049

    Chicago Manual of Style (17th edition)

osu1525455392722049
273
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This open access ETD is published by The Ohio State University and OhioLINK.