So you want to get a PhD..
Published:
while I think this is a great idea, let me tell you: This is not a degree for the faint-hearted.
Posts by Tags
PhD
categories
Recursion as an Effect
Published:
In one of my previous post I showed that any theory featuring general recursion is inconsistent when viewed as a logical system which inevitably leads to the idea that all definable functions in such a theory should be total (or productive).
On Lax Monoidal Functors
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What is the difference between
- a lax monoidal functor
- a monoid in a Day-monoidal category
- a morphism of lax-algebras for the free monoid 2-monad, and
- a codistributive law with the tensor product?
Bisimulations, Equality and Traces
Published:
Strong bisimulation for CCS is the preferred equivalence method in concurrency because it relates less programs than trace equality. However, the reality is that is strong bisimulation and trace equality ought to be regarded as equivalent. This is the essence behind proof assistant’s like (e.g.) Isabelle. So what is going here?
The mini Yoneda lemma for Type Theorists
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I have managed to teach the Yoneda lemma to students who knew very little about category theory, here’s how you do it.
CCCs and the complete models of STLC
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Cartesian closed categories are not regarded as complete models of the Simply Typed \(\lambda\)-calculus in the traditional sense. Let’s see why.
Inconsistencies in Cartesian Closed Categories with fixed-points
Published:
Any Cartesian Closed Category (CCC) with an initial object and a fixed-point operator is trivial. Essentially this means that in languages like (e.g.) Haskell the empty type is not actually empty as it contains the non-terminating computation. Perhaps this is obvious, but here’s the categorical explanation.
domain-theory
Recursion as an Effect
Published:
In one of my previous post I showed that any theory featuring general recursion is inconsistent when viewed as a logical system which inevitably leads to the idea that all definable functions in such a theory should be total (or productive).
foundations
The Axiom of Choice in Type Theory
Published:
The Axiom of Choice (AC) is an axiom that states that the product of a family of non-empty sets is itself non-empty. This is a rather controversial axiom amongst mathematicians but in type theory this axiom is provable within the logic.
monads
Recursion as an Effect
Published:
In one of my previous post I showed that any theory featuring general recursion is inconsistent when viewed as a logical system which inevitably leads to the idea that all definable functions in such a theory should be total (or productive).
On Lax Monoidal Functors
Published:
What is the difference between
- a lax monoidal functor
- a monoid in a Day-monoidal category
- a morphism of lax-algebras for the free monoid 2-monad, and
- a codistributive law with the tensor product?
monoids
On Lax Monoidal Functors
Published:
What is the difference between
- a lax monoidal functor
- a monoid in a Day-monoidal category
- a morphism of lax-algebras for the free monoid 2-monad, and
- a codistributive law with the tensor product?
recursion
Recursion as an Effect
Published:
In one of my previous post I showed that any theory featuring general recursion is inconsistent when viewed as a logical system which inevitably leads to the idea that all definable functions in such a theory should be total (or productive).
Inconsistencies in Cartesian Closed Categories with fixed-points
Published:
Any Cartesian Closed Category (CCC) with an initial object and a fixed-point operator is trivial. Essentially this means that in languages like (e.g.) Haskell the empty type is not actually empty as it contains the non-terminating computation. Perhaps this is obvious, but here’s the categorical explanation.
semantics
Recursion as an Effect
Published:
In one of my previous post I showed that any theory featuring general recursion is inconsistent when viewed as a logical system which inevitably leads to the idea that all definable functions in such a theory should be total (or productive).
Bisimulations, Equality and Traces
Published:
Strong bisimulation for CCS is the preferred equivalence method in concurrency because it relates less programs than trace equality. However, the reality is that is strong bisimulation and trace equality ought to be regarded as equivalent. This is the essence behind proof assistant’s like (e.g.) Isabelle. So what is going here?
The mini Yoneda lemma for Type Theorists
Published:
I have managed to teach the Yoneda lemma to students who knew very little about category theory, here’s how you do it.
CCCs and the complete models of STLC
Published:
Cartesian closed categories are not regarded as complete models of the Simply Typed \(\lambda\)-calculus in the traditional sense. Let’s see why.
Inconsistencies in Cartesian Closed Categories with fixed-points
Published:
Any Cartesian Closed Category (CCC) with an initial object and a fixed-point operator is trivial. Essentially this means that in languages like (e.g.) Haskell the empty type is not actually empty as it contains the non-terminating computation. Perhaps this is obvious, but here’s the categorical explanation.
set theory
The Axiom of Choice in Type Theory
Published:
The Axiom of Choice (AC) is an axiom that states that the product of a family of non-empty sets is itself non-empty. This is a rather controversial axiom amongst mathematicians but in type theory this axiom is provable within the logic.
stlc
CCCs and the complete models of STLC
Published:
Cartesian closed categories are not regarded as complete models of the Simply Typed \(\lambda\)-calculus in the traditional sense. Let’s see why.