Orthomodularity of Decompositions in a Categorical Setting

2006 ◽  
Vol 45 (6) ◽  
pp. 1079-1090 ◽  
Author(s):  
John Harding
Keyword(s):  
Categorical Setting

On the Foundation of Combinatorial Theory. X. A Categorical Setting for Symmetric Functions

1992 ◽  
Vol 86 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Flavio Bonetti ◽  
Gian-Carlo Rota ◽  
Domenico Senato ◽  
Antonietta M. Venezia
Keyword(s):  
Symmetric Functions ◽  
Combinatorial Theory ◽  
Categorical Setting ◽  
Theory X

Algebras for combinatorial search

2009 ◽  
Vol 19 (3-4) ◽  
pp. 469-487 ◽  
Author(s):  
J. MICHAEL SPIVEY
Keyword(s):  
Search Strategies ◽  
Algebraic Specification ◽  
Combinatorial Search ◽  
Algebraic Specifications ◽  
Categorical Setting

AbstractCombinatorial search strategies including depth-first, breadth-first and depth-bounded search are shown to be different implementations of a common algebraic specification that emphasizes the compositionality of the strategies. This specification is placed in a categorical setting that combines algebraic specifications and monads.

scholarly journals Completeness types for uniformity theory on textures

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 159-178 ◽  
Author(s):  
Filiz Yıldız
Keyword(s):  
Uniform Spaces ◽  
Categorical Setting

In this paper the author considers the various types of completeness for di-uniform texture spaces and especially for complemented ones. Following that, the relationships between completeness of uniform spaces and these types of completeness for complemented di-uniform texture spaces are investigated in a categorical setting, just as interrelations between quasi-uniform spaces and di-uniform texture spaces are pointed out insofar as completeness is concerned. Additionally, useful requirements among the various types of completeness of a di-uniformity and real dicompactness of the uniform ditopological space generated by that di-uniformity are presented as a diagram.

CATEGORICAL HARMONY AND PATH INDUCTION

2017 ◽  
Vol 10 (2) ◽  
pp. 301-321 ◽  
Author(s):  
PATRICK WALSH
Keyword(s):  
Recent Work ◽  
Homotopy Type ◽  
Category Theory ◽  
Type Theory ◽  
Homotopy Type Theory ◽  
Categorical Setting ◽  
Philosophical Framework

AbstractThis paper responds to recent work in the philosophy of Homotopy Type Theory by James Ladyman and Stuart Presnell. They consider one of the rules for identity, path induction, and justify it along ‘pre-mathematical’ lines. I give an alternate justification based on the philosophical framework of inferentialism. Accordingly, I construct a notion of harmony that allows the inferentialist to say when a connective or concept is meaning-bearing and this conception unifies most of the prominent conceptions of harmony through category theory. This categorical harmony is stated in terms of adjoints and says that any concept definable by iterated adjoints from general categorical operations is harmonious. Moreover, it has been shown that identity in a categorical setting is determined by an adjoint in the relevant way. Furthermore, path induction as a rule comes from this definition. Thus we arrive at an account of how path induction, as a rule of inference governing identity, can be justified on mathematically motivated grounds.

scholarly journals Towards a typed Geometry of Interaction

2010 ◽  
Vol 20 (3) ◽  
pp. 473-521 ◽  
Author(s):  
ESFANDIAR HAGHVERDI ◽  
PHILIP SCOTT
Keyword(s):  
Linear Logic ◽  
Independent Interest ◽  
Cut Elimination ◽  
Mathematical Framework ◽  
Monoidal Categories ◽  
Original Theory ◽  
Categorical Setting ◽  
Abstract Notion ◽  
The Relationship

Girard's Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cut elimination. We introduce a typed version of GoI, called Multiobject GoI for both multiplicative linear logic (MLL) and multiplicative exponential linear logic (MELL) with units. We present a categorical setting that includes our previous (untyped) GoI models, as well as more general models based on monoidal *-categories. Our development of multiobject GoI depends on a new theory of partial traces and trace classes, which we believe is of independent interest, as well as an abstract notion of orthogonality (which is related to work of Hyland and Schalk). We develop Girard's original theory of types, data and algorithms in our setting, and show his execution formula to be an invariant of cut elimination (under some restrictions). We prove soundness theorems for the MGoI interpretation (for Multiplicative and Multiplicative Exponential Linear Logic) in partially traced *-categories with an orthogonality. Finally, we briefly discuss the relationship between our GoI interpretation and other categorical interpretations of GoI.

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