17 citations · 40 across the 13 of their papers we have counts for
14 papers
Rank-based linkage I: triplet comparisons and oriented simplicial complexes
R. W. R. Darling, Will Grilliette, Adam Logan
Rank-based linkage is a new tool for summarizing a collection of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between o…
Simplification for Graph-like Objects
Will Grilliette
The simplification of a multigraph into a simple graph can be abstracted to a more general comma category under some common conditions. When using the identity functor, the categor…
Incidence Hypergraphs: Box Products & the Laplacian
Will Grilliette, Lucas J. Rusnak
The box product and its associated box exponential are characterized for the categories of quivers (directed graphs), multigraphs, set system hypergraphs, and incidence hypergraphs…
Incidence Hypergraphs: Injectivity, Uniformity, and Matrix-tree Theorems
Will Grilliette, Josephine Reynes, Lucas J. Rusnak
An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic subs…
Incidence hypergraphs: The categorical inconsistency of set-systems and a characterization of quiver exponentials
Will Grilliette, Lucas J. Rusnak
This paper considers the difficulty in the set-system approach to generalizing graph theory. These difficulties arise categorically as the category of set-system hypergraphs is sho…
A Functorial Link between Quivers and Hypergraphs
Will Grilliette
This paper discusses some issues arising from the category of hypergraphs, the category of (undirected) multigraphs, and the topos of q…
Scaled-free objects II
Will Grilliette
This work creates two categories of "array-weighted sets" for the purposes of constructing universal matrix-normed spaces and algebras. These universal objects have the analogous u…
Matricial Banach spaces
Will Grilliette
This work performs a study of the category of complete matrix-normed spaces, called matricial Banach spaces. Many of the usual constructions of Banach spaces extend in a natural wa…
On Blow-Ups and Injectivity of Quivers
Will Grilliette, Deborah E. Seacrest, Tyler Seacrest
This work connects the idea of a "blow-up" of a quiver with that of injectivity, showing that for a class of monic maps , a quiver is -injective if and only if all blow-ups o…
Injective Envelopes and Projective Covers of Quivers
Will Grilliette
This paper characterizes the injective and projective objects in the category of directed multigraphs, or quivers. Further, the injective envelope and projective cover of any quive…
Universal Properties of Some Quivers
Will Grilliette
In this paper, I characterize four particular classes of directed multigraphs, or quivers, as images under left and right adjoints to the natural vertex and edge functors. In parti…
The C*-algebra of a Single Invertible Element
Will Grilliette
This paper characterizes the unital C*-algebra generated by a single invertible element as the unital free product of C[0,1] and C(T). To do this, I develop techniques to split and…
Presentations and Tietze transformations of C*-algebras
Will Grilliette
In this work, I develop a new view of presentation theory for C*-algebras, both unital and non-unital, heavily grounded in classical notions from algebra. In particular, I introduc…
Scaled-Free Objects
Will Grilliette
In this work, I address a primary issue with adapting categorical and algebraic concepts to functional analytic settings, the lack of free objects. Using a "normed set" and associa…