When An Abelian Category Has Enough Flat Objects?

Is there a category theory of models that would unite theories from different academic disciplines?

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What does it mean to unify in this context? Are you aware that different sciences use different methods and because of that use different mathematics, and even in mathematics itself category theory has limited applications? That said, there are several branches outside pure mathematics where category theory found its applications. For example take theoretical CS, where it's used in several places, for example in domain theory (which aims at providing mathematical foundations to one approach to semantics of programming languages). It also helps understanding Curry-Howard isomorphism (or maybe rather is part of the correspondence). Some wizards from UK even made IO for purely functional programming language Haskell using monads, which come from category theory (this I think can be thought of as a part of algebraification programme).

Kuba Bartczuk

Though  Category Theory has a reputation as difficult (and too abstract) there  is some movement towards exploring it as a unifying framework for  understanding the relationships between different theoretical areas that  currently have their own home-grown mathematics.  This would start out  by exploring functional analogies but might one day become a common  mathematical foundation.  As a non-mathematician I wonder whether this  could provide a basic theory of models in place of the current use of  set theory.  The following article explores this idea: http://scholar.google.com/scholar?q=related%3APgH55oF8WggJ%3Ascholar.google.com%2F&hl=en&as_sdt=0%2C48&as_ylo=2005&as_yhi=2015 In  physics, Feynman diagrams are used to reason about quantum processes.   In the 1980s, it became clear that underlying these diagrams is a  powerful analogy between quantum physics and topology.  Namely, a linear  operator behaves very much like a ‘cobordism’:  a manifold representing  spacetime, going between two manifolds representing space.   This led  to a burst of work on topological  quantum  field theory and ‘quantum  topology’.  But this was just the beginning:  similar diagrams can be  used to reason about logic, where they represent proofs, and  computation, where they represent programs.  With the rise of interest  in quantum cryptography and quantum computation, it became clear that  there is extensive network of analogies between physics,  topology,   logic and computation.   In this expository paper,  we make some of  these analogies precise using the concept of ‘closed symmetric monoidal  category’.  We assume no prior knowledge of category theory, proof  theory or computer science. ... By  now there is an extensive network of interlocking analogies between  physics, topology, logic and computer science. They suggest that  research in the area of common overlap is actually trying to build a new  science:  a general science of systems and processes. Building this  science will be very difficult. There are good reasons for this, but  also bad ones. One bad reason is that different fields use different  terminology and notation. The  original Rosetta Stone, created in 196 BC, contains versions of the  same text in three languages:  demotic Egyptian,  hieroglyphic script  and classical  Greek.   Its rediscovery by Napoleon’s soldiers let  modern Egyptologists decipher the hieroglyphs. Eventually this led to a  vast increase in our understanding of Egyptian culture. At  present, the deductive systems in mathematical logic look like  hieroglyphs to most physicists. Similarly, quantum field theory is Greek  to most computer scientists, and so on. So, there is a need for a new  Rosetta Stone to aid researchers attempting to translate between fields.

Jeff Wright

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