14
votes
1answer
602 views

Lawvere's fixed point theorem and the Recursion Theorem

Building off of Qiaochu's comment on my answer to a previous mathoverflow question, I would like to know: can the Recursion Theorem, $$\forall e\exists k[\Phi_e\text{ is total }\implies ...
7
votes
2answers
764 views

Categories of recursive functions

I have a couple of conjectures on recursive functions, that I feel must have been proved or refuted by someone else, but I don't know where to look. In short: 1. The primitive recursive functions ...
9
votes
3answers
731 views

Effective topos and computability in topological spaces

The classical computability theory taking place in $\mathbb{N}$, can be extended to more general spaces, like $T_0$ second countable topological spaces $(X, \mathcal{O}, v)$ where $\mathcal{O}$ is a ...
2
votes
0answers
203 views

A polytime feasible subuniverse of the Effective Topos

The effective topos is a well known universe of sets suitable for abstract computability, as it is build "from the ground up" via the classical notion of realisability by Kleene. I have found a few ...
13
votes
2answers
631 views

Are the axioms for higher category-theory effectively computable?

I ask this, although I don't conduct any research in the area, or even plan to. -- There seems to be general agreement that the axioms for higher categories grow very rapidly in complexity as the ...
1
vote
2answers
391 views

Pullbacks for primitive recursive functions.

Since a pullback of two functions f and g with common codomain into Set category is just a subset of cartesian product like this: {(x,y)/f(x)=g(y)} (with two more functions not important here) could ...
19
votes
3answers
2k views

Is the theory of categories decidable?

There are a lot of theorems in basic homological algebra, such as the five lemma or the snake lemma, that seem like they'd be more easily proven by computer than by hand. This led me to consider the ...
1
vote
1answer
480 views

Coequalizer in the category of primitive recursive functions

What does a coequalizer in the category of primitive recursive functions look like? I know that in Set, a coequalizer is a minimum congruence but...what is it in particular in the category of ...