### Sheaves, Cosheaves and Applications

, 2013

"... This note advertises the theory of cellular sheaves and cosheaves, which are devices for conducting linear algebra parametrized by a cell complex. The theory is presented in a way that is meant to be read and appreciated by a broad audience, including those who hope to use the theory in applications ..."

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This note advertises the theory of cellular sheaves and cosheaves, which are devices for conducting linear algebra parametrized by a cell complex. The theory is presented in a way that is meant to be read and appreciated by a broad audience, including those who hope to use the theory in applications across science and engineering disciplines. We relay two approaches to cellular cosheaves. One relies on heavy

### Higher Dimensional Categories: Model Categories and Weak Factorisation Systems

, 2007

"... Loosely speaking, “homotopy theory ” is a perspective which treats objects as equivalent if they have the same “shape ” which, for a category theorist, occurs when there exists a certain class W of morphisms that one would like to invert, but which are not in fact isomorphisms. Model categories prov ..."

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Loosely speaking, “homotopy theory ” is a perspective which treats objects as equivalent if they have the same “shape ” which, for a category theorist, occurs when there exists a certain class W of morphisms that one would like to invert, but which are not in fact isomorphisms. Model categories provide a setting in which one can do “abstract homotopy

### Reoccurring Patterns in Hierarchical Protein Materials and Music: The Power of Analogies

, 2011

"... Abstract Complex hierarchical structures composed of simple nanoscale building blocks form the basis of most biological materials. Here, we demonstrate how analogies between seemingly different fields enable the understanding of general principles by which functional properties in hierarchical syste ..."

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Abstract Complex hierarchical structures composed of simple nanoscale building blocks form the basis of most biological materials. Here, we demonstrate how analogies between seemingly different fields enable the understanding of general principles by which functional properties in hierarchical systems emerge, similar to an analogy learning process. Specifically, natural hierarchical materials like spider silk exhibit properties comparable to classical music in terms of their hierarchical structure and function. As a comparative tool, here, we apply hierarchical ontology logs that follow a rigorous mathematical formulation based on category theory to provide an insightful system representation by expressing knowledge in a conceptual map. We

### On the Topology of Discrete Planning with Uncertainty

, 2011

"... This paper explores the topology of planning with uncertainty in discrete spaces. The paper defines strategy complex as the collection of all plans for accomplishing all tasks specified by goal states in a finite discrete graph. Transitions in the graph may be nondeterministic or stochastic. One key ..."

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This paper explores the topology of planning with uncertainty in discrete spaces. The paper defines strategy complex as the collection of all plans for accomplishing all tasks specified by goal states in a finite discrete graph. Transitions in the graph may be nondeterministic or stochastic. One key result is that a system can attain any state in its graph despite control uncertainty if and only if its strategy complex is homotopic to a sphere of dimension two less than the number of states in the graph.

### FUNCTIONAL PEARL Data types àlacarte

"... This paper describes a technique for assembling both data types and functions from isolated individual components. We also explore how the same technology can be used to combine free monads and, as a result, structure Haskell’s monolithic IO monad. 1 ..."

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This paper describes a technique for assembling both data types and functions from isolated individual components. We also explore how the same technology can be used to combine free monads and, as a result, structure Haskell’s monolithic IO monad. 1

### SUMMARY Contents

"... D2.2.1 Application of models of cooperation to network operation, design of P2P application, and social research through ..."

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D2.2.1 Application of models of cooperation to network operation, design of P2P application, and social research through

### Generalized Topological Semantics for First-Order Modal Logic

, 2010

"... This dissertation provides a new semantics for first-order modal logic. It is philosophically motivated by the epistemic reading of modal operators and, in particular, three desiderata in the analysis of epistemic modalities. (i) The semantic modelling of epistemic modalities, in particular verifia ..."

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This dissertation provides a new semantics for first-order modal logic. It is philosophically motivated by the epistemic reading of modal operators and, in particular, three desiderata in the analysis of epistemic modalities. (i) The semantic modelling of epistemic modalities, in particular verifiability and falsifiability, cannot be properly achieved by Kripke’s relational notion of accessibility. It requires instead a more general, topological notion of accessibility. (ii) Also, the epistemic reading of modal operators seems to require that we combine modal logic with fully classical first-order logic. For this purpose, however, Kripke’s semantics for quantified modal logic is inadequate; its logic is free logic as opposed to classical logic. (iii) More importantly, Kripke’s semantics comes with a restriction that is too strong to let us semantically express, for instance, that the identity of Hesperus and Phosphorus, even if metaphysically necessary, can still be a matter of epistemic discovery. To provide a semantics that accommodates the three desiderata, I show, on the one hand, how the desideratum (i) can be achieved with topological semantics, and more generally neighborhood semantics, for propositional modal logic. On the other hand, to achieve (ii) and (iii), it turns out

### Categorial compositionality continued: A category theory explanation for quasi-systematicity

"... universal construction The classical account for systematicity of human cognition supposes: (1) syntactically compositional representations; and (2) processes that are sensitive to their structure. The problem with this account is that there is no explanation as to why these two components must be c ..."

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universal construction The classical account for systematicity of human cognition supposes: (1) syntactically compositional representations; and (2) processes that are sensitive to their structure. The problem with this account is that there is no explanation as to why these two components must be compatible, other than by ad hoc assumption (convention) to exclude nonsystematic variants that, e.g., mix prefix and postfix concatenative compositional schemes. Recently, we proposed an alternative explanation (Phillips & Wilson, 2010) without ad hoc assumptions, using a branch of mathematics, called category theory. In this paper, we extend our explanation to domains that are quasi-systematic (e.g., language), where the domain includes some but not all possible combinations of constituents. The central category-theoretic construct is an adjunction involving pullbacks, where the focus is on the relations between processes, rather than the representations. In so far as cognition is systematic, the basic building blocks of cognitive architecture are adjunctions by our theory.

### An Exposition of Sheaves

, 2009

"... This survey serves to introduce sheaves and some basic properties of sheaves from a categorical perspective. The objective is that if one has a basic understanding of category theory that this approach to defining sheaves is more intuitive than the usual approach of a typical Algebraic Geometry text ..."

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This survey serves to introduce sheaves and some basic properties of sheaves from a categorical perspective. The objective is that if one has a basic understanding of category theory that this approach to defining sheaves is more intuitive than the usual approach of a typical Algebraic Geometry text. We