Foundational to the subject of operad theory is the notion of an En operad, that is, an operad that is quasi-isomorphic to the operad of little n-cubes Cn. They are central to the study of iterated loop spaces, and the specific case of n = 2 is key in the solution of the Deligne Conjecture. In this paper we examine the connection between two E 2 operads, namely the little 2-cubes operad C 2 itself and the operad of spineless cacti. To this end, we construct a new suboperad of C2, which we name the operad of tethered 2-cubes. Much of our analysis involves examining trees labeled by elements of the operad of little intervals, C1. In the final chapter, we generalize this idea of graphs decorated by elements of an operad to the notion of a decorated Feynman category, building off of the work of Kaufmann and Ward. As an immediate application, we will give a simple definition of non-Σ modular operads.
","abstract_html":"<p>Foundational to the subject of operad theory is the notion of an <em>En</em> operad, that is, an operad that is quasi-isomorphic to the operad of little <em>n-</em>cubes <em>Cn.</em> They are central to the study of iterated loop spaces, and the specific case of <em>n</em> = 2 is key in the solution of the Deligne Conjecture. In this paper we examine the connection between two <em>E</em> 2 operads, namely the little 2-cubes operad <em>C</em> 2 itself and the operad of spineless cacti. To this end, we construct a new suboperad of <em>C</em>2, which we name the operad of tethered 2-cubes. Much of our analysis involves examining trees labeled by elements of the operad of little intervals, <em>C</em>1. In the final chapter, we generalize this idea of graphs decorated by elements of an operad to the notion of a decorated Feynman category, building off of the work of Kaufmann and Ward. As an immediate application, we will give a simple definition of non-Σ modular operads.</p>","abstract_has_math":false,"creators":["Lucas, Jason M"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ralph Kaufmann","James McClure","David B. McReynolds","Jeremy Miller"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-08-01T07:00:00Z","date_published":"2016-08-01T07:00:00Z","updated_at":"2026-07-24T03:54:02Z","subjects":["Pure sciences","Algebraic topology","Category theory","Operad theory","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://docs.lib.purdue.edu/open_access_dissertations/802","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ralph Kaufmann","James McClure","David B. McReynolds","Jeremy Miller"]},{"key":"dc:creator","label":"Author","values":["Lucas, Jason M"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Pure sciences","Algebraic topology","Category theory","Operad theory","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://docs.lib.purdue.edu/open_access_dissertations/802"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Foundational to the subject of operad theory is the notion of an En operad, that is, an operad that is quasi-isomorphic to the operad of little n-cubes Cn. They are central to the study of iterated loop spaces, and the specific case of n = 2 is key in the solution of the Deligne Conjecture. In this paper we examine the connection between two E 2 operads, namely the little 2-cubes operad C 2 itself and the operad of spineless cacti. To this end, we construct a new suboperad of C2, which we name the operad of tethered 2-cubes. Much of our analysis involves examining trees labeled by elements of the operad of little intervals, C1. In the final chapter, we generalize this idea of graphs decorated by elements of an operad to the notion of a decorated Feynman category, building off of the work of Kaufmann and Ward. As an immediate application, we will give a simple definition of non-Σ modular operads.
"]},{"key":"dc:title","label":"Title","values":["Connecting models of configuration spaces: From double loops to strings"]}]}],"canonical_facts":{"dc:contributor":["Ralph Kaufmann","James McClure","David B. McReynolds","Jeremy Miller"],"dc:creator":["Lucas, Jason M"],"dc:description.abstract":["Foundational to the subject of operad theory is the notion of an En operad, that is, an operad that is quasi-isomorphic to the operad of little n-cubes Cn. They are central to the study of iterated loop spaces, and the specific case of n = 2 is key in the solution of the Deligne Conjecture. In this paper we examine the connection between two E 2 operads, namely the little 2-cubes operad C 2 itself and the operad of spineless cacti. To this end, we construct a new suboperad of C2, which we name the operad of tethered 2-cubes. Much of our analysis involves examining trees labeled by elements of the operad of little intervals, C1. In the final chapter, we generalize this idea of graphs decorated by elements of an operad to the notion of a decorated Feynman category, building off of the work of Kaufmann and Ward. As an immediate application, we will give a simple definition of non-Σ modular operads.
"],"dc:identifier":["https://docs.lib.purdue.edu/open_access_dissertations/802"],"dc:subject":["Pure sciences","Algebraic topology","Category theory","Operad theory","Mathematics"],"dc:title":["Connecting models of configuration spaces: From double loops to strings"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:54:02Z"}