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Dec
11
comment Verifying that a function is a morphism by checking a generating set
This question only makes sense for concrete categories, and very concrete ones at that: these generating sets are really just sets and not objects in the same category.
Dec
11
comment Is the halting of a program that checks for duplicates in an infinite multiset decidable?
Yes, but even then there are still problems, as you pointed out in your answer. I was trying to suggest that the naive interpretation of the question is uninteresting: as soon as you allow inputs of infinite length, then any program that checks for duplicates in the input would have to run forever, and so it fails to halt in a very trivial way.
Dec
11
comment Inductive vs projective limit of sequence of split surjections
I don't see why not. Instead of taking the whole of $\mathbb{Z}^{\times \mathbb{N}}$, we could take the submodule generated by $\mathbb{Z}^{\oplus \mathbb{N}}$ (which is countable) and the sequence $(1, 1, 1, \ldots)$; in other words, this is the space of eventually-constant sequences.
Dec
11
comment What is the difference between necessary and sufficient conditions?
Those are the definitions of necessary and of sufficient.
Dec
11
answered Automorphism of an elementary extension of a structure that moves an undefinable element
Dec
11
comment Is Hom$(G,-)$ left exact if morphisms are required to be continuous?
$\textrm{Hom}(-, G)$ is very rarely right exact, but again, if it fails to be left exact, you have defined exact sequence incorrectly.
Dec
11
awarded  algebraic-geometry
Dec
10
comment Is the halting of a program that checks for duplicates in an infinite multiset decidable?
How do you code $\Sigma$? In the standard definition of computability, inputs to programs are required to be coded as natural numbers – in particular, you can't code the entire powerset of an infinite set.
Dec
10
comment Is Hom$(G,-)$ left exact if morphisms are required to be continuous?
If $\textrm{Hom}(G, -)$ fails to be left exact then you have defined exact sequence incorrectly.
Dec
10
comment Non-Free Finitely Generated Injective Modules over a Local Ring
Let $k$ be a field. Then $(k, (0))$ is a local ring, and $k$ is a finitely-generated injective (and projective and free!) $k$-module.
Dec
10
comment Model existence for infinitary logics
Syntactic, of course. There are obvious infintary analogues of the usual rules of inference.
Dec
10
comment Types and Algebraicity
What's your definition of algebraic type? This is the definition I'm familiar with...
Dec
10
awarded  Nice Answer
Dec
10
comment Winding number of algebraic curves
Hmmm. What applications of winding number are you hoping to capture here? Perhaps thinking about those will lead to an idea for a good definition, in the usual French style of turning theorems into definitions...
Dec
10
answered Good references for stacks
Dec
9
answered Why must we distinguish between rational and irrational numbers?
Dec
9
comment Why must we distinguish between rational and irrational numbers?
Start with nothing (a.k.a. $0$), and keep adding one...
Dec
9
comment Prove that these two objects are isomorphic
@sunflower Yes: the Yoneda lemma implies that the functor $A \mapsto \mathcal{C}(-, A)$ is a fully faithful embedding of $\mathcal{C}$ into $[\mathcal{C}^\textrm{op}, \textbf{Set}]$, and any fully faithful functor reflects isomorphisms.
Dec
9
comment Inductive vs projective limit of sequence of split surjections
I'm afraid I can't think of any reason why that should be the case, nor any counterexample.
Dec
9
revised Inductive vs projective limit of sequence of split surjections
added 141 characters in body