Loop Spaces and Iterated Higher Dimensional Enrichment
Forcey, Stefan Andrew
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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.
- Doctoral Dissertations