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May
4
comment Covariant and Contravariant Functor of Fixed Set Question - Category of Sets
The question is imprecise. After all, the formulae $X \mapsto X^A$ and $X \mapsto A^X$ do not specify what happens to morphisms. The point is to define actual functors whose object parts are as specified – so there's really nothing much to it at all.
May
2
comment Classifying space infinite totally ordered set contractible
@NajibIdrissi $\omega_1$ has a bottom element, so its nerve is contractible (in the strong sense). More generally, any category with an initial object has contractible nerve.
May
1
comment Adjunction between topological and simplicial presheaf categories
$\mathcal{C}^\Delta$ is not good notation, so I will write $\mathcal{D}$ instead. It is not hard to see that $\mathbf{Top} \to \mathbf{sSet}$ induces a functor sending topologically enriched functors $\mathcal{C} \to \mathbf{Top}$ to simplicially enriched functors $\mathcal{D} \to \mathbf{sSet}$, but I don't see any candidate for a left adjoint. (Actually, the only functors going the other way that I can think of are either constants or factor through the homotopy category.)
May
1
comment Tor Functor Commutes with Direct Limits
Classically, "direct limit" means "colimit of a directed diagram". In particular, they are filtered colimits.
Apr
30
comment Can I define a predicate over a set and use it in the definition of another one?
Are you trying to work in a specific axiomatic set theory?
Apr
30
comment How to understand cocategories
For any $\mathcal{C}$ with pullbacks, cocategories in $\mathcal{C}$ are the same as functors $\mathcal{C} \to \mathbf{Cat}$ whose composites with $\operatorname{ob}, \operatorname{mor} : \mathbf{Cat} \to \mathbf{Set}$ are representable. Such functors are not so common, but there are a few examples, e.g. the left and right adjoints of $\operatorname{ob} : \mathbf{Cat} \to \mathbf{Set}$.
Apr
30
comment Characterization of epic morphisms in the category of rings.
See this question.
Apr
30
comment Points of scheme with residue field $k$ vs $k$-point
That's not possible when all the morphisms are $k$-morphisms.
Apr
29
comment Choice of a skeleton
Actually, in my mind, a skeleton is not just a full subcategory but also equipped with a quasi-inverse to the inclusion. At any rate, all you need is a sufficiently strong axiom of choice (and the law of excluded middle).
Apr
29
comment Is $S^1 \times S^1$ really a torus?
The torus has many (Riemannian) metrics. One of them is even flat!
Apr
28
comment Positive and negative logical connectives
If you could add more details about the translation of intuitionistic logic into linear logic and explain the positive/negative distinction in linear logic, that would be much appreciated.
Apr
28
comment Do finite products commute with colimits in the category of spaces?
On (3): Well, it suffices for $(-) \times X$ to have a right adjoint, so you can assume $X$ is locally compact Hausdorff.
Apr
28
comment Category of $\mathcal{L}$-structures
My guess is that $\mathbf{Struct}_\mathcal{L}$ is locally finitely presentable.
Apr
27
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.
Apr
27
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.
Apr
26
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.
Apr
26
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 $\mathbf{Top}$.
Apr
26
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.
Apr
26
comment $\mathsf{Top}$ with proper maps has products.
That's obviously false: the category of spaces + proper maps has no terminal object.
Apr
26
comment $\mathsf{Top}$ with proper maps has products.
Surely it must be finite. The infinite version implies Tychonoff's theorem.