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Apr
21
revised Does the functor $\mathbf{cosk_n}:sSet\to sSet$ preserve Kan complexes?
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Apr
21
awarded  Nice Answer
Apr
18
comment (Non-)Isomorphism of (pre-)sheaves
It is true for sheaves and not true for presheaves. That is, in some the sense, the whole point about sheaves.
Apr
17
comment Categorical Banach space theory
I don't see why the category of normed vector spaces (or rather their unit balls) should be locally finitely presentable. There's no obvious essentially algebraic axiomatisation.
Apr
17
answered What does it mean for pullbacks to preserve monomorphisms?
Apr
17
comment Are planes without $n$ points isomorphic as algebraic varieties for different n?
The automorphism group of $\mathbb{A}^1$ acts 2-transitively, so you can remove any two points you like. I don't know whether you can remove any three points you like, though.
Apr
17
comment Definition of a regular category via extremal epi
If you look at his definition of "cover" you will see that it is what everyone else calls "extremal epimorphism". The only hypothesis needed is that covers are epimorphisms.
Apr
17
comment Is every monomorphism a homonomorphism?
In the derived category of chain complexes, the homotopy kernel is zero if and only if the morphism is a quasi-isomorphism.
Apr
16
comment Definition of a regular category via extremal epi
Look in the section on regular categories in Sketches of an elephant.
Apr
16
comment What does it mean for pullbacks to preserve monomorphisms?
Correct. Actually, you can conclude $p_A$ is a monomorphism if $f_B$ is.
Apr
16
comment Looking for info on power set functor
Not especially. You can start here.
Apr
16
comment Is the product of all objects of a finite category an initial object?
@armchairprogrammer Perhaps the result you are thinking of is that if the limit of the identity diagram exists, then it is an initial object.
Apr
16
comment Reference request for the independence of $ \text{Con}(\mathsf{ZFC}) $.
If ZFC is $\omega$-consistent, then $\lnot \mathrm{Con}(\mathrm{ZFC})$ is not provable. (If it were, then by $\omega$-consistency, ZFC is inconsistent – a contradiction!)
Apr
16
comment Correct definition of subframe
That is correct.
Apr
16
comment Looking for info on power set functor
For better or worse, that's what it's called here as well.
Apr
15
comment Looking for info on power set functor
The observation you speak of is called Frobenius reciprocity: $\exists_f (X' \cap f^{-1} Y') = \exists_f X' \cap Y'$. It is equivalent to the fact that $f^{-1}$ preserves the Heyting implication.
Apr
15
comment Looking for info on power set functor
These are not functors between categories but rather monotone maps between posets.
Apr
15
comment Looking for info on power set functor
$\exists_f$ is the left adjoint of pullback $f^{-1}$, $\forall_f$ is the right adjoint of $f^{-1}$.
Apr
15
comment Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?
The $X$-th power. Or the category of functors $X \to \mathbb{B} \mathrm{Aut} (A)$.
Apr
15
comment In Ring Theory, does a 'power' of a morphism represent composition?
You can't make the set of continuous functions $\mathbb{R} \to \mathbb{R}$ into a ring with pointwise addition and composition. You would have to restrict to linear functions, in which case you get something isomorphic to $\mathbb{R}$ again.