Title page for ETD etd-04232004-160123


Type of Document Dissertation
Author Forcey, Stefan Andrew
Author's Email Address sforcey@math.vt.edu
URN etd-04232004-160123
Title Loop Spaces and Iterated Higher Dimensional Enrichment
Degree PhD
Department Mathematical Physics
Advisory Committee
Advisor Name Title
Quinn, Frank S. Committee Chair
Floyd, William J. Committee Member
Green, Edward L. Committee Member
Hagedorn, George A. Committee Member
Haskell, Peter E. Committee Member
Keywords
  • n-categories
  • enriched categories
  • iterated loop spaces
  • iterated monoidal categories
Date of Defense 2004-04-01
Availability unrestricted
Abstract
This thesis pursues the hints of a categorical delooping that are suggested when enrichment is iterated. At each stage of successive enrichments, the number of monoidal products decreases and the categorical dimension increases, both by one. This is mirrored by topology. When we consider the loop space of a topological space, we see that paths (or 1--cells) in the original are now points (or objects) in the derived space. There is also automatically a product structure on the points in the derived space, where multiplication is given by concatenation of loops. Delooping is the inverse functor here, and thus involves shifting objects to the status of 1--cells and decreasing the number of ways to multiply.

We define V-(n+1)--categories as categories enriched over V--n--Cat, the (k-n)--fold monoidal strict (n+1)--category of V--n--categories. We show that for V k--fold monoidal the structure of a (k-n)--fold monoidal strict (n+1)--category is possessed by V--n--Cat.

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