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Aug
20
comment A question on a property of geometric morphisms related to locales.
Not at all. They mean what they say.
Aug
20
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
Ah, yes – you're right.
Aug
19
comment Is there any example of a Category without generators?
Indeed. So coproducts + zero object is enough.
Aug
18
comment Translation of 'morphisme net'?
It would be good if you could give the definition (in translation).
Aug
18
comment If weak-equivalences in a model category are generated by an equivalence relation $\sim$ on hom-sets…
I doubt bifibrancy is enough (or even necessary). You should think about how the proof works for the case $\mathcal{C} = \mathbf{Top}$ and the classical model structure – then it will be clear that you should ask for some notion of path object or cylinder object with respect to $\sim$.
Aug
17
comment Cech Cohomology and the Dold-Kan Correspondence
The simplicial identities force them to be (generalised) diagonal embeddings. Use the universal property of projections.
Aug
17
comment Cech Cohomology and the Dold-Kan Correspondence
If you have an open cover you can get a simplicial object.
Aug
16
comment Is the Quillen Injective Model Structure on the category of positive cochain complexes of R-modules (co)fibrantly generated?
I think the answer is yes, but it may be difficult to get an explicit generating set of trivial cofibrations. The key phrase is "Smith's theorem".
Aug
16
comment Intuition for homotopy (co)limits in triangulated categories
The analogue to think about is an abelian category with countable coproducts. How would you construct colimits for sequences using use coproducts and cokernels?
Aug
15
comment Left adjoint of the forgetful functor $\mathsf{Grpd} \to \mathsf{Cat}$?
Yes, the left adjoint exists. The morphisms are zigzags, as you say.
Aug
15
comment Is the reflective localization $L_WC$ of a category $C$ equivalent to $C$? What am I missing?
Sometimes functors are not the obvious ones.
Aug
14
comment Why is $\mathsf{Type} : \mathsf{Type}$ a contradiction?
No, it is not a matter of notation. Girard's paradox is more like the Burali-Forti paradox.
Aug
14
comment Replacing a covering in a site with a single arrow
The answer is essentially contained here
Aug
14
comment Is any splitting field algebraic?
Actually, $K[X] / (f)$ may not be a splitting field for $f$.
Aug
14
comment Why is $\mathsf{Type} : \mathsf{Type}$ a contradiction?
@user40276 No. This is Girard's paradox.
Aug
13
comment Is the parametrization $(t^3,t^6)$, a reparametrization of $(t,t^2)$?
Well, then you don't have a reparametrisation, because the inverse is not smooth.
Aug
13
comment Is the parametrization $(t^3,t^6)$, a reparametrization of $(t,t^2)$?
Obviously that theorem depends on the definition of "reparametrisation".
Aug
12
comment degree of an etale cover of the affine line
If $X \to \mathbb{A}^1_k$ is a finite étale cover and $X$ is connected then it is an isomorphism. No? For $k = \mathbb{C}$ this corresponds to the fact that $\mathbb{C}$ is simply connected.
Aug
11
comment Choosing projective replacement to be functorial
1. Classical derived functors are functors – between the derived categories. 2. Look up Cartan–Eilenberg resolutions. 3. A functorial projective replacement should be enough. 4. There is only one possible meaning.
Aug
11
comment Can you integrate on a scheme?
Integrate what?