Date of Award

Summer 2022

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Maria Basterra

Second Advisor

Donald Hadwin

Third Advisor

Edward Hinson

Abstract

An operad can be thought of as a collection of operations, each with a finite number of inputs and a single output, along with a composition rule. We prove that the category of operads in an appropriate concrete symmetric monoidal category V is equivalent to a subcategory of symmetric monoidal categories enriched in V. Though versions of this result have appeared previously in the literature, we prove that a more restrictive subcategory is needed to construct the equivalence. Our subcategory has the advantage that its objects share important properties with the historical precursor to operads, PROPs.

We also review a localization construction for operads, called the tree hammock localization. Using the above equivalence, we compare this construction to the hammock localization for categories. We believe that these two localization constructions should be suitably equivalent, and present ongoing work on this conjecture using simplicial categories and ∞-categories.

Download

Share

COinS