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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.)
Jun
25
comment Problem based category-theory book.
Perhaps you can try taking a classic (e.g. Categories for the working mathematician) and try to prove all the theorems yourself.
Jun
24
answered base change of an equivalence relation of fppf sheaves
Jun
24
comment base change of an equivalence relation of fppf sheaves
There's a minor subtlety in that $G \to Q$ needs to be the coequaliser in the category of presheaves, not sheaves. But otherwise that's the idea.
Jun
24
comment base change of an equivalence relation of fppf sheaves
It is true that epimorphisms are preserved by base change, but the technique I described is not directly applicable. Try instead to show that the coequaliser of a given pair of morphisms is preserved by base change (in the appropriate sense).
Jun
24
comment base change of an equivalence relation of fppf sheaves
There's nothing special about fppf sheaves here. These are standard facts about all sheaf toposes, codified in Giraud's axioms. A cheap proof is to observe that the statements in question are true in $\mathbf{Set}$ and sheafification preserves the truth of these statements.
Jun
24
awarded  Popular Question
Jun
23
comment On Levy's formal definition of class terms
You don't really "apply" it. This is an example of mutual recursion.
Jun
23
comment Singleton Sets Not Open
This seems to be an unusual property – the only discrete space with this property is empty...
Jun
23
comment Why is the full subcategory consisting of simply connected spaces not complete?
It seems to me to be difficult to relate limits of simply connected pointed spaces with limits of their underlying sets. For one thing, the underlying set functor is not representable.
Jun
23
comment On Levy's formal definition of class terms
Sure. But there are more complicated things: see e.g. here.
Jun
23
comment On Levy's formal definition of class terms
This is indeed a definition by simultaneous recursion. Have you seen other examples of that?
Jun
23
comment Category of sets and multi-valued functions
Indeed, the OP's description of the category is exactly the definition of the Kleisli category for the powerset monad.
Jun
23
answered What's so special about the grounded poset of cardinality $2$?
Jun
22
comment Ring structure of a localization
It's defined like in the field of fractions. Do you know how that is done?