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Apr
26
comment $\mathsf{Top}$ with proper maps has products.
Surely it must be finite. The infinite version implies Tychonoff's theorem.
Apr
23
comment Positive and negative logical connectives
I did have the sense that I might have reversed the labels. Interesting!
Apr
23
comment A localization of a topos is still a topos
Right. Well, as I said, it's not an easy result. Look up universal closure operators, e.g. [Sheaves in geometry and logic, Ch. V §1] or [Sketches of an elephant, Part A, §4.3].
Apr
22
comment Definiton of Limit and Foundational problems.
@user233650 Well, if you care enough to formalise things in set theory, then you must already secretly believe that every "object" is a set, even if not every collection is a set.
Apr
22
comment Definiton of Limit and Foundational problems.
Or, if you want to discuss a specific large category, you can use unbounded quantifiers – just like in the axioms of set theory.
Apr
21
comment A localization of a topos is still a topos
What do you mean by "localisation"? What is true is this: if $\mathcal{E}$ is a topos and $\mathcal{E}'$ is a full subcategory of $\mathcal{E}$ such that the inclusion $\mathcal{E}' \hookrightarrow \mathcal{E}$ has a left adjoint that preserves finite limits, then $\mathcal{E}'$ is also a topos. It is relatively straightforward to show that $\mathcal{E}'$ is cartesian closed – the hard part is constructing the subobject classifier.
Apr
21
comment If $V$ is a vector space $\neq$ the vector space of its additive identity alone, must $V$ have a subspace $\neq V$?
Well, after the 0-dimensional vector space, what's the next smallest vector space?
Apr
21
comment Does the functor $\mathbf{cosk_n}:sSet\to sSet$ preserve Kan complexes?
If you examine the argument above, you will see that $\mathbf{cosk}_n$ preserves Kan fibrations whose codomain has the right lifting property with respect to the inclusion $\partial \Delta^{n+1} \hookrightarrow \Delta^{n+1}$. For a Kan complex, this is equivalent to the condition that $\pi_n$ is trivial. In particular, $\mathbf{cosk}_0$ preserves Kan fibrations whose codomain is a connected Kan complex.
Apr
18
comment (Non-)Isomorphism of (pre-)sheaves
It is true for sheaves and not true for presheaves. That is, in some the sense, the whole point about sheaves.
Apr
17
comment Categorical Banach space theory
I don't see why the category of normed vector spaces (or rather their unit balls) should be locally finitely presentable. There's no obvious essentially algebraic axiomatisation.
Apr
17
comment Are planes without $n$ points isomorphic as algebraic varieties for different n?
The automorphism group of $\mathbb{A}^1$ acts 2-transitively, so you can remove any two points you like. I don't know whether you can remove any three points you like, though.
Apr
17
comment Definition of a regular category via extremal epi
If you look at his definition of "cover" you will see that it is what everyone else calls "extremal epimorphism". The only hypothesis needed is that covers are epimorphisms.
Apr
17
comment Is every monomorphism a homonomorphism?
In the derived category of chain complexes, the homotopy kernel is zero if and only if the morphism is a quasi-isomorphism.
Apr
16
comment Definition of a regular category via extremal epi
Look in the section on regular categories in Sketches of an elephant.
Apr
16
comment What does it mean for pullbacks to preserve monomorphisms?
Correct. Actually, you can conclude $p_A$ is a monomorphism if $f_B$ is.
Apr
16
comment Looking for info on power set functor
Not especially. You can start here.
Apr
16
comment Is the product of all objects of a finite category an initial object?
@armchairprogrammer Perhaps the result you are thinking of is that if the limit of the identity diagram exists, then it is an initial object.
Apr
16
comment Reference request for the independence of $ \text{Con}(\mathsf{ZFC}) $.
If ZFC is $\omega$-consistent, then $\lnot \mathrm{Con}(\mathrm{ZFC})$ is not provable. (If it were, then by $\omega$-consistency, ZFC is inconsistent – a contradiction!)
Apr
16
comment Correct definition of subframe
That is correct.
Apr
16
comment Looking for info on power set functor
For better or worse, that's what it's called here as well.