Dissertation
The cell categories Theta_n: dualities, traces, and coherent nerves
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2026
DOI: 10.25820/etd.008354
Abstract
This thesis is a three-part study of the cell categories $\Theta_n$ and their uses in higher category theory.
In the first part, we generalize the known duality between $\Theta_n$ and Joyal's categories of finite combinatorial $n$-disks by producing a functorial construction of Joyal's disks which is naturally dual to the categorical wreath product which produces $\Theta_n$. This allows us to describe the duals of other categorical wreath products. We explore applications to crossed simplicial groups.
In the second part, we generalize the cyclic nerve and trace of a 1-category to the context of strict $n$-categories. In particular, for any $n$-category $C$ and each $n\geq 0$ and $0\leq i<n$ we define a higher cyclic nerve $N^{\cyc_{n,i}}C$ and trace $\tr^{n,i}C$. We explore several characterizations of these constructions. We prove that $\tr^{n,n-1}C$ recovers a type of trace already present in the literature.
In the third part, we obtain for all $n\geq 1$ a right Quillen nerve $N^\frak{h}:n\cat{-Cat}\to \psh(\Theta_n)$ where $\psh(\Theta_n)$ carries Ara's model structure. This functor is a machine which inputs $n$-categories and outputs $(\infty, n)$-categories. We additional prove that, as expected, a map of $n$-quasi-categories is an equivalence precisely when it is fully faithful and essentially surjective.
Details
- Title: Subtitle
- The cell categories Theta_n: dualities, traces, and coherent nerves
- Creators
- Nicholas Cecil
- Contributors
- Ben Cooper (Advisor)Frauke Bleher (Committee Member)Charles Frohman (Committee Member)Ryan Kinser (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2026
- DOI
- 10.25820/etd.008354
- Publisher
- University of Iowa
- Number of pages
- ix, 124 pages
- Copyright
- Copyright 2026 Nicholas Cecil
- Language
- English
- Date submitted
- 04/26/2026
- Description illustrations
- illustrations
- Description bibliographic
- Includes bibliographical references (pages 122-124).
- Public Abstract (ETD)
Mathematicians study the relationships between mathematical objects. This can be straightforward (e.g. all squares are rectangles) or more complex (e.g. the ways of walking between two points in some space). In these more complex cases, it is productive to view the relationships (e.g. the paths) as mathematical objects in their own right. Since relationships can be chained together as a sort of multiplication, this leads to an algebra of relationships which is called category theory. The primary object of study is called a category: a collection of objects and a network of relationships between them.
If we study the relationships between mathematical objects, and the relationships are mathematical objects themselves, it is natural to study relationships between relationships (abbreviated as 2-relationships). The study of these notions is called 2-category theory, the primary object of study: 2-categories.
The game does not stop here. There are 3-categories, 4-categories, and so on. At the highest level, there are ∞-categories which have objects, relationships, and relationships between relationships, relationships between these, and so on without end.
There is a family Θ1, Θ2, Θ3, ... of very simple categories which are used to describe the theory of n-categories (n = 1, 2, 3, ...,∞). This work is a study of these categories Θn and some applications to n-category theory. In particular, we address some technical details at the heart of defining Θn (which lead to some new mathematical objects of interest) and build several new operations on n-categories which we proceed to study.
- Academic Unit
- Mathematics
- Record Identifier
- 9985177074502771
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