Starting from the four normed division algebras---the real numbers, complex

numbers, quaternions and octonions, with dimensions k=1, 2, 4 and 8,

respectively---a systematic procedure gives a 3-cocycle on the Poincar'e Lie

superalgebra in dimensions k+2=3, 4, 6 and 10. A related procedure gives a

4-cocycle on the Poincare Lie superalgebra in dimensions k+3=4, 5, 7 and

11. The existence of these cocycles follow from certain spinor identities that

hold only in these dimensions, and which are closely related to the existence

of superstring and super-Yang--Mills theory in dimensions k+2, and

super-2-brane theory in dimensions k+3.

In general, an (n+1)-cocycle on a Lie superalgebra yields a `Lie

n-superalgebra': that is, roughly speaking, an n-term chain complex

equipped with a bracket satisfying the axioms of a Lie superalgebra up to chain

homotopy. We thus obtain Lie 2-superalgebras extending the Poincare

superalgebra in dimensions 3, 4, 6, and 10, and Lie 3-superalgebras extending

the Poincar'e superalgebra in dimensions 4, 5, 7 and 11. As shown in Sati,

Schreiber and Stasheff's work on generalized connections valued in Lie

n-superalgebras, Lie 2-superalgebra connections describe the parallel

transport of strings, while Lie 3-superalgebra connections describe the

parallel transport of 2-branes. Moreover, in the octonionic case, these

connections concisely summarize the fields appearing in 10- and 11-dimensional

supergravity.

Generically, integrating a Lie n-superalgebra to a Lie n-supergroup yields a

`Lie n-supergroup' that is hugely infinite-dimensional. However,

when the Lie n-superalgebra is obtained from an (n+1)-cocycle on a

nilpotent Lie superalgebra, there is a geometric procedure to integrate the

cocycle to one on the corresponding nilpotent Lie supergroup.

In general, a smooth (n+1)-cocycle on a supergroup yields a `Lie

n-supergroup': that is, a weak n-group internal to supermanifolds. Using

our geometric procedure to integrate the 3-cocycle in dimensions 3, 4, 6 and

10, we obtain a Lie 2-supergroup extending the Poincare supergroup in those

dimensions, and similarly integrating the 4-cocycle in dimensions 4, 5, 7 and

11, we obtain a Lie 3-supergroup extending the Poincar'e supergroup in those

dimensions.

Fong has developed a compositional framework by the name of \emph{decorated cospans} that is well-suited for modeling `open' networks and systems. In this framework, open networks are seen as the morphisms of a category and can be composed as such allowing larger open networks to be built up from smaller ones. Much work has already been done in this direction, and is where the story of this dissertation starts. During the process of my first work to promote Fong's theory of decorated cospan categories to decorated cospan \emph{bicategories}, it was noticed that the morphisms which made up these decorated cospan categories had a certain unexpected and stringent characterization to them. From this observation, one of the main frameworks of the present thesis, namely `structured cospans', was conceived by John Baez. The main version of the Structured cospans framework utilizes `double categories' which are similar in flavor to bicategories in that they have a 2-dimensional structure to them. Shortly thereafter, we decided to revisit Fong's decorated cospan machinery also from the perspective of double categories to improve the original decorated cospans framework. Much work in this thesis is built on the foundations of double categories, and while working with both the structured cospans and decorated cospans frameworks, a double category of open Markov processes was also built in which `coarse-grainings' appear as 2-morphisms and are shown to be compatible with `black-boxing'.

This thesis aims to develop a compositional theory for the operational semantics of networks. The networks considered are described by either internal or enriched graphs. In the internal case we focus on $\mathsf{Q}$-nets, a generalization of Petri nets based on a Lawvere theory $\mathsf{Q}$. $\mathsf{Q}$-nets include many known variants of Petri nets including pre-nets, integer nets, elementary net systems, and bounded nets. In the enriched case we focus on graphs enriched in a quantale $R$ regarded as matrices with entries in $R$. These $R$-matrices represent distance networks, Markov processes, capacity networks, non-deterministic finite automata, simple graphs, and more. The operational semantics of $\mathsf{Q}$-nets is constructed as an adjunction between $\mathsf{Q}$-nets and categories internal to the category of models of $\mathsf{Q}$. The left adjoint of this adjunction sends a $\mathsf{Q}$-net $P$ to an internal category $F_{\mathsf{Q}}(P)$ whose morphisms represent all possible firing sequences in $P$. Similarly, the operational semantics of $R$-matrices is constructed as an adjunction between $R$-matrices and categories enriched in $R$. The left adjoint of this adjunction sends an $R$-matrix $M$ to the $R$-category $F_R(M)$ whose hom-objects are solutions of the algebraic path problem: a generalization of the shortest path problem to graphs weighted in $R$. For both $\mathsf{Q}$-nets and $R$-matrices we use the theory of structured cospans to study the compositionality of the above operational semantics. For each type of network we construct a double category whose morphisms are ``open networks", i.e.\ networks with certain vertices designated as input or output. The operational semantics gives a double functor from a double category of open networks to a double category of open enriched or internal categories. These double functors give a compositional framework for computing the operational semantics of $\mathsf{Q}$-nets and $R$-matrices: their functoriality and coherence give relationships between the operational semantics of a network and the operational semantics of the smaller networks from which it is composed. We introduce the black-boxing of an open network, a profunctor describing the externally observable behavior of an open network. We introduce a class of open networks called ``functional open networks" for which black-boxing preserves composition.

This thesis develops the foundations of the program of groupoidification and presents an application of this program --- the Fundamental Theorem of Hecke Operators. In stating this theorem, we develop a theory of enriched bicategories and construct the Hecke bicategory --- a categorification of the intertwining operators between permutation representations of a finite group. As an immediate corollary, we obtain a categorification of the Iwahori-Hecke algebra, which leads to solutions of the Zamolodchikov tetrahedron equation. Such solutions are a positive step towards invariants of 2-tangles in 4-dimensional space and constructions of higher-categories with braided structures.