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6h
comment Definition of adjoint functor and locally small categories
Well, obviously. But my point is that this is just the direct translation of the hom-set definition into the language of two-sided fibrations.
16h
comment Definition of adjoint functor and locally small categories
This is actually the same as the hom-set definition. The fibres of the the two functors are the hom-sets, and the fact that $\phi$ is a functor corresponds to naturality of the bijection.
1d
revised Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way?
edited tags
1d
comment Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way?
possible duplicate of Categorical Pasting Lemma
1d
comment $\mathsf{Top}$ with proper maps has products.
No. The point is that products in the category of spaces + continuous maps are not necessarily products in the category of spaces + proper maps.
1d
comment $\mathsf{Top}$ with proper maps has products.
No, the correct statement is that the class of proper maps is closed under finite products in $\mathbf{Top}$.
1d
comment $\mathsf{Top}$ with proper maps has products.
@EricAuld A category with finite products in particular has an empty product, i.e. a terminal object.
1d
comment $\mathsf{Top}$ with proper maps has products.
That's obviously false: the category of spaces + proper maps has no terminal object.
1d
comment $\mathsf{Top}$ with proper maps has products.
Surely it must be finite. The infinite version implies Tychonoff's theorem.
Apr
24
answered The class of all functions between classes (NBG)
Apr
23
comment Positive and negative logical connectives
I did have the sense that I might have reversed the labels. Interesting!
Apr
23
asked Positive and negative logical connectives
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
21
revised Does the functor $\mathbf{cosk_n}:sSet\to sSet$ preserve Kan complexes?
added 106 characters in body
Apr
21
awarded  Nice Answer