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Introduction: Paul Cadman

Hi,

I'm a software engineer in London interested in using functional programming and type theory to make software better (for a nice definition of better).

Before I got into software development I completed a PhD in Algebraic Geometry. Category Theory was very useful in my studies and research so I was thrilled to see that the same ideas have applications in software engineering. I'm excited to explore this more with everyone.

Comments

  • 1.

    Hi! What sort of algebraic geometry did you, and how was category theory involved? I have a hobby of slowly learning bits of algebraic geometry.

    It's cool that topoi, first developed by Grothendieck for algebraic geometry, wound up becoming a new approach to logic and are now being used by David Spivak and others to do things like design air traffic control systems.

    Comment Source:Hi! What sort of algebraic geometry did you, and how was category theory involved? I have a hobby of slowly learning bits of algebraic geometry. It's cool that topoi, first developed by Grothendieck for algebraic geometry, wound up becoming a new approach to logic and are now being used by David Spivak and others to do things like design air traffic control systems.
  • 2.

    Hi John, thanks for starting this group.

    Yes - it's wonderful that those new approaches are being exploited from some very abstract and general theories.

    My work was in singularity theory - specifically studying shapes representing deformations of other shapes containing singularities. Category theory is used in many different ways, as you know it's very versatile. It's significance in maths goes very deep and I can't claim to understand more than a very small part of it.

    Here's a couple of more practical ways category theory was useful for me.

    Firstly it's used to give short and precise definitions and statements. It's a very good guide if your definition comes from a universal property of a category rather than some ad-hoc construction. It's more likely to be useful in different contexts (for example). It also eases communication. I could tell a colleague that a property 'was functorial' (i.e was consistent with the functor in the context of our discussion) and they would know what I mean precisely without distracting explanations.

    A large fraction of singularity theory (and algebraic geometry) is expressed in terms of homological algebra. Here you translate features of geometric objects into infinite sequences of algebraic objects related by arrows. In these terms the properties and invariants of the geometric objects fall out as categorical constructions (functors: Ext, Tor, Hom for example).

    For example we may want to consider deformations of a shape that preserve some property. For complex shapes in high dimensions we don't have much intuition about what nice deformations would be. However the 'niceness' of a deformation is linked to 'flatness' which in turn can be precisely defined using the Tor functor. In this way we can build up new intuition about geometry that is better suited to the problems we're working on.

    Comment Source:Hi John, thanks for starting this group. Yes - it's wonderful that those new approaches are being exploited from some very abstract and general theories. My work was in singularity theory - specifically studying shapes representing deformations of other shapes containing singularities. Category theory is used in many different ways, as you know it's very versatile. It's significance in maths goes very deep and I can't claim to understand more than a very small part of it. Here's a couple of more practical ways category theory was useful for me. Firstly it's used to give short and precise definitions and statements. It's a very good guide if your definition comes from a universal property of a category rather than some ad-hoc construction. It's more likely to be useful in different contexts (for example). It also eases communication. I could tell a colleague that a property 'was functorial' (i.e was consistent with the functor in the context of our discussion) and they would know what I mean precisely without distracting explanations. A large fraction of singularity theory (and algebraic geometry) is expressed in terms of homological algebra. Here you translate features of geometric objects into infinite sequences of algebraic objects related by arrows. In these terms the properties and invariants of the geometric objects fall out as categorical constructions (functors: Ext, Tor, Hom for example). For example we may want to consider deformations of a shape that preserve some property. For complex shapes in high dimensions we don't have much intuition about what nice deformations would be. However the 'niceness' of a deformation is linked to 'flatness' which in turn can be precisely defined using the Tor functor. In this way we can build up new intuition about geometry that is better suited to the problems we're working on.
  • 3.

    Are you playing around with possible relation between SemiAlgRel and Corr_\mathbb{C} ?

    Comment Source:Are you playing around with possible relation between SemiAlgRel and Corr_\mathbb{C} ?
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