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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
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
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 $\left| I \right| = 2$.
Dec
27
comment $k$-point after base change
What's your definition of variety?
Dec
27
comment Categorical Pasting Lemma
@Exterior It is true in the special case you consider. For the $n > 2$ case you need a very different diagram.
Dec
26
comment What are coquantales?
A frame is a monoid in $\mathbf{Sup}$, and the coproduct in $\mathbf{Frm}$ is the tensor product in $\mathbf{Sup}$.
Dec
26
comment What are coquantales?
For instance, any localic group has an underlying Hopf algebra.
Dec
25
comment What's the intution behind defining the cotangent sheaf as $\Delta^\ast(\mathscr{I}/\mathscr{I}^2)$?
Well, another way of looking at it is this: $Z$ is Hausdorff/separated if and only if, for every parallel pair $f_0, f_1 : Y \to Z$, the equaliser of $f_0$ and $f_1$ is a closed subspace/subscheme of $Y$.
Dec
25
comment Query in the definion of abelian category
@egreg Only if you assume Hausdorffness.
Dec
24
comment Understanding the (categorical) Calculus of Fractions
The point, as it were, is to have Propositions 5.2.4 and 5.2.5.
Dec
22
comment Functorial Properties Preserved by Natural Isomorphism
So you are claiming that elementarily equivalent groups of the same cardinality are isomorphic. This is also false. (Take an ultrapower of $\mathbb{Z}$ and use Löwenheim–Skolem to extract a countable elementary submodel that is non-cyclic.)
Dec
22
comment Functorial Properties Preserved by Natural Isomorphism
Your claim is incorrect: you are saying two elementarily equivalent groups are isomorphic, but this is false for infinite groups.
Dec
21
comment The uniqueness principle for products in the HoTT Book
It has to do with the induction principle for products.
Dec
21
comment The uniqueness principle for products in the HoTT Book
I would not say that the judgemental equality for $x$ follows from the judgemental equality for $(a, b)$, but certainly the propositional equality follows.