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1d
comment Intuitionistic Proof of $(a \Rightarrow b) \Rightarrow (\lnot b \Rightarrow \lnot a)$
@z5h No, the point is that $\lnot a$ is an abbreviation of $a \to \bot$, so if you want to prove $\lnot a$, you should assume $a$ and prove $\bot$; similarly, $\lnot b$ is an abbreviation of $b \to \bot$, so if you want to deduce $\bot$ from $\lnot b$, you should prove $b$.
1d
comment Distributivity of pullbacks
This is essentially the definition of a lextensive category.
1d
comment Axiomatizability of the algebra of (a fragment of) calculus
I believe the answer is yes, because $S$ contains $\mathbb{Z} [x]$, and it seems to me that $\mathbb{Z} [x]$ is the free commutative ring equipped with a derivation $D$ and an element $x$ such that $D x = 1$.
1d
comment Why is a simply connected 3-manifold a homotopy 3-sphere?
You should also include the part where triangularisability and Poincaré duality are explained.
2d
comment Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$
Pointed topological spaces.
2d
comment Properties preserved under equivalence of categories
Everything in that list is preserved by either a left adjoint or a right adjoint, and equivalences are simultaneously left and right adjoints.
Jul
1
comment Is there a special name for functors from a category C to a subcategory of C?
They are endofunctors.
Jul
1
comment Group action on a category
As I said, it is a special case of a pseudofunctor.
Jul
1
comment Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$
They are different. You should clarify which one you mean.
Jun
29
comment Group action on a category
Yes, it is a pseudofunctor.
Jun
29
comment How to show that a map is finite
You could use the valuative criterion for properness, but I think it would be much easier to just use the standard definition of finite morphism in terms of finite algebras.
Jun
29
comment How to show that a map is finite
Well, quasi-finiteness is clear, right? So you just need to check that the map is proper.
Jun
29
comment How to show that a map is finite
What definition are you using?
Jun
28
comment when will homology and direct limit commute?
There seem to be two questions here: whether homology commutes with direct limits, and how the homology of the mapping telescope is related to the direct limit of the homology.
Jun
27
comment Representable morphism for algebraic spaces
We identify schemes with the corresponding representable sheaves.
Jun
27
comment Equivalence of group objects in set and groups as one object categories.
(1) You can define the notion of a category object and what it means to have only one object and every morphism is an isomorphism. (2) Yes. (3) Yes.
Jun
26
comment Product of Schemes and Open Subsets
It does not follow from the universal property. You have to show that the product of open immersions is an open immersion; it follows from the fact that open immersions are closed under base change and composition, but that's still something that needs to be checked.
Jun
26
comment “World's Hardest Easy Geometry Problem”
That's not what I mean. If you actually use the geometry of the plane then there is no problem.
Jun
26
comment Product of Schemes and Open Subsets
As Hoot says, it is essentially part of the construction of fibred products. You should review that.
Jun
25
comment Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$
I use the classical definition of $\mathbf{hTop}_*$ – the morphisms are based homotopy classes of based maps. So the isomorphisms are based homotopy equivalences. (Model structures are not the only way of defining homotopy categories.)