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Aug
25
comment When does Sheafification commute with direct image?
There are already counterexamples to your suggestions in the answer.
Aug
24
comment Question about “immediate” observation about finitely presentable objects
The quotient of a congruence is literally a quotient in the classical sense, and the kernel pair of any homomorphism is a congruence. So the coequaliser has to be the quotient by the smallest congruence. Think concretely!
Aug
24
comment How to give “categorical” specifications of categories like Grp?
Yes, but that doesn't preclude stupid descriptions, e.g. a generators-and-relations presentation...
Aug
24
revised When does Sheafification commute with direct image?
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Aug
24
answered When does Sheafification commute with direct image?
Aug
24
comment Can we bypass connection?
Connections fail to be tensors because they measure the difference between a tensor and a non-tensor. So they are non-tensors by design.
Aug
24
comment Is it possible to define Cauchy sequences in a topological space?
Also, the property of being a complete metric space is not invariant under homeomorphism.
Aug
24
comment About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$
It is usually considered bad form to ask lots of questions at once.
Aug
23
comment Functor preserves kernels iff it's left exact
Actually, the functor has to preserve $0$ and $\oplus$ as well.
Aug
23
comment Reference for a couple of terms, $\underline{\operatorname{Hom}}_X(-,-)$ and $\boxtimes$
The first one is the hom sheaf, the second one is the external tensor product.
Aug
21
comment A question on a property of geometric morphisms related to locales.
A localic geometric morphism is not literally a locale, as I explained. But there is a correspondence. A localic geometric morphism $\mathcal{E} \to \mathcal{S}$ is proper if and only if the corresponding internal locale in $\mathcal{S}$ is proper. But one still has to define what it means for an internal locale in $\mathcal{S}$ to be proper.
Aug
20
awarded  Popular Question
Aug
20
comment Composition of Dual Maps in (rigid) Monoidal Categories
You should always be able to translate string diagram manipulations into equations. If you can't, perhaps you should start by reviewing that.
Aug
20
revised Composition of Dual Maps in (rigid) Monoidal Categories
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Aug
20
comment A question on a property of geometric morphisms related to locales.
I don't know any good reference. What you have to understand first is Grothendieck's relative point of view: a localic geometric morphism $\mathcal{E} \to \mathcal{S}$ is something like "a locale from the point of view of $\mathcal{S}$". Indeed, the 2-category of localic geometric morphisms $\mathcal{E} \to \mathbf{Set}$ (for variable $\mathcal{E}$) is equivalent to the 2-category of locales. But for general $\mathcal{S}$ one has to speak of "internal locales in $\mathcal{S}$".
Aug
20
comment A question on a property of geometric morphisms related to locales.
$P$ is not supposed to be a property of morphisms of locales; rather $P$ is a property of localic geometric morphisms or locales. I think you need to review the definitions.
Aug
20
comment A question on a property of geometric morphisms related to locales.
$L$ itself is not a localic geometric morphism (or a locale), so how could it possibly have property $P$?
Aug
20
comment A question on a property of geometric morphisms related to locales.
Not at all. They mean what they say.
Aug
20
answered A question on a property of geometric morphisms related to locales.
Aug
20
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
Ah, yes – you're right.