Dear Steve, Thanks for taking the interest in the paper. Whether or not the colimit exists depends on which category you are in. In my paper the colimit is taken in the category of small categories where it does exist. Like you point out, (and David Roberts pointed out in a previous post) you might want to do some of this stuff in other categories such as the effective topos etc where this colimit would not exist. I do not know if the category of small categories is "omniscient" or not, and if that is a good thing or a bad thing, but the result stands. (It is not a typo. Since 2 is indiscreet 0-->1 once the output goes to 1 it must stay in 1. This is exactly what Halt(x,y,t) does. Once it halts it stays halted. But all the outputs can remain 0.) All the best, Noson -----Original Message----- From: Steve Vickers [mailto:s.j.vickers@cs.bham.ac.uk] Sent: Monday, September 21, 2015 6:10 AM To: Noson S. Yanofsky Cc: categories@mta.ca Subject: Re: categories: Computability and Complexity of Categorical Structures Dear Noson, It seems to me it's an oversimplification to say that categories can solve the Halting Problem. There's a logical principle involved. In your proof on p.23, the central step is that if H: ω -> 2 is a functor (where ω and 2 are the posets of natural numbers and {0,1}), then it has a colimit H': 1 -> 2. H(t) describes whether the computation has halted by step t. (By the way, you use ω_i, the indiscreet natural numbers, for the type of t. That's a typo, isn't it? Since 2 is antisymmetric, that would imply H is constant.) Does the colimit exist? That's not a fact of pure category theory, but a logical principle of "omniscience". If you go beyond classical logic (and category theory facilitates this) then the colimit might not exist. For example, it will not work in a topos. To find the colimit you will need to embed 2 in Ω, the poset on the subobject classifier. There you still have the classical objects 0 and 1 (false and true), but also intermediate objects corresponding to truth values of other propositions such as "this computation halts". This switch from 2 to Ω corresponds to relaxing decidability to semidecidability - and semidecidability of the halting problem is trivial. If my calculations are correct, you can see this in the topos of sheaves over the 1-point compactification of ω. This is the classifying topos for functors H, and the generic H has no colimit. An alternative way to understand this is what you see in denotational semantics. Here, when a map to 2 takes the value 0 ("bottom"), that is interpreted as meaning that the assignment of result to argument gets ensnared in a divergence. Regards, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]