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Jan
1
comment How to determine whether a differential $1$-form is globally welldefined?
What are $d x$ and $d y$?
Jan
1
comment What is $\rightarrowtail$ used for?
Actually, you are supposed to use $\mapsto$ here.
Jan
1
comment Explicitly describe colimits in $\mathsf{Set}$
Sure, that's one way of looking at it.
Jan
1
answered Explicitly describe colimits in $\mathsf{Set}$
Jan
1
comment Explicitly describe colimits in $\mathsf{Set}$
Not everything can be dualised. After all, $\mathbf{Set}^\mathrm{op}$ is very far from $\mathbf{Set}$.
Dec
31
answered “Twist” of $\mathbb P^n_K$ through a field automorphism.
Dec
30
comment Killing homotopy groups
Well, $i_* (\alpha) = [i \circ f]$ and $i \circ f$ factors through a contractible space...
Dec
30
comment True or false? If $\eta$ is an explicitly defined incomputable number, then no formal system can pin down the value $\eta$ to arbitrary precision.
So, you have an algorithm that determines what ZFC proves or not?
Dec
30
comment Direct way to show that 2-out-of-6 holds for weak equivalences in a model category?
It's certainly true that in actual examples we often get 2-out-of-6 or even saturation for free. That's why [Dwyer, Hirschhorn, Kan, and Smith] advocate changing the definition.
Dec
30
comment Direct way to show that 2-out-of-6 holds for weak equivalences in a model category?
No, that's not true. In order to construct the model structure on $\mathbf{Top}$ you need to do almost all the hard work needed to prove Whitehead's theorem anyway. At least that's what Moerdijk remarked. There's never any free lunch.
Dec
30
comment Direct way to show that 2-out-of-6 holds for weak equivalences in a model category?
I don't think there's any easy way to do this. If you can show that homotopy equivalences are weak equivalences then you are already halfway to showing the class of weak equivalences is saturated.
Dec
30
comment sheaves of rings and maps to classifying topos
Yes, it is important that $\mathcal{R}$ is the category of finitely presented commutative rings. Yes, it is relevant that $\mathcal{R}$ is generated under colimits by $\mathbb{Z}[x]$, but it is not true that $\mathcal{R}$ is freely generated. You might like to think about what sheaf of rings corresponds to the identity geometric morphism – this will be the universal example.
Dec
29
answered Discrete simplicial spaces are fibrant
Dec
29
comment On the definition of a Type
"Type" can be defined to the same degree of formality as "set". So you might ask yourself first, what is a set?
Dec
29
awarded  Yearling
Dec
29
comment Yoneda Embedding into Left Exact Functors
@Hurkyl You mean, limits and colimits in $[\mathcal{A}, \mathbf{Set}]$ can be computed pointwise. But anything can happen in a subcategory.
Dec
28
comment Does a $p$-form eat $p$-vectors or $p$ number of vectors?
Either. Both. After all, you can take $p$ vectors and wedge them together to get a $p$-vector.
Dec
27
comment $k$-point after base change
It feels to me that smoothness is a rather strong condition. I would have guessed that being geometrically integral is enough.
Dec
27
comment Categorical Pasting Lemma
Just write out what it means.
Dec
27
comment Categorical Pasting Lemma
$\varinjlim$ simply denotes the colimit. Your claim is the case where $\left| I \right| = 2$.