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Mar
6
comment Tangent spaces of compact spaces
It is just definitions. Compactness has no relation to tangent spaces. (Note, however, that the tangent space of a point is $0$-dimensional, hence is compact.)
Mar
6
comment Is there a categorical construction of the general linear group?
The ring of $n \times n$ matrices over $A$, of course.
Mar
6
comment How many Riemann surfaces homeomorphic to the sphere are there?
The sphere is a surface of genus $0$, so the Hurwitz formula implies there can be no unramified coverings of it by a compact surface of positive genus. Hence the uniformisation theorem implies any Riemann surface that is topologically a sphere must be $\mathbb{C P}^1$.
Mar
6
comment Kleisli category examples
You can always form the Kleisli or the Eilenberg–Moore category to get an adjunction. In short, every monad comes from an adjunction.
Mar
5
comment When are zero morphisms preserved?
If you have a functor between two categories with a zero object, then the functor preserves zero morphisms if and only if it preserves the zero object. But if you can't even prove that in your example then you probably can't say much...
Mar
5
comment What is the term for a graph on $n$ vertices with no edges?
I would call it a discrete graph, but I am not a graph theorist.
Mar
5
comment Relate two descriptions of the stack $[X/G]$
I'm not entirely convinced that the first description is literally correct. It certainly should be the case that $[X/G]$ is a classifying stack for some kind of $G$-bundle, with $X \to [X/G]$ being the universal such, but it's not obvious to me exactly what conditions are needed. Also, the associated stack ("stackification") may change the objects and morphisms in the fibres.
Mar
5
comment Will $i^*$ pull back injectives to injectives?
Exact functors preserve finite products.
Mar
4
comment An existence of a right Kan extension
I would add the hypothesis of local-smallness. It's not hard to dream up an example where the category $(c \downarrow K)$ is large.
Mar
4
comment How to demystify the axioms of propositional logic?
I wouldn't say that everything provable constructively is provable using $S$ and $K$ alone. In constructive logic one also needs to take $\lor$ and $\land$ as primitive, and accordingly, one needs more axioms.
Mar
3
comment How many Riemann surfaces homeomorphic to the sphere are there?
Alternatively, use the uniformisation theorem and Riemann–Hurwitz.
Mar
3
comment A real number being computable
How do you know when to stop?
Mar
2
comment Relate two descriptions of the stack $[X/G]$
For the second construction, where it says "point", read "scheme morphism with codomain $X$". In other words, define $[X / G]$ to be the stack associated with the (strict!) functor $\textbf{Sch}^\textrm{op} \to \textbf{Grpd}$ that sends $T$ to the groupoid whose objects are $T$-valued points of $X$ and whose morphisms are $g : x \to x'$ where $g \in G(T)$ and $x' = g \cdot x$.
Mar
2
comment what is the definition of $=$?
Your proposal to define $A = B$ as $P (A) \leftrightarrow P (B)$ for all predicates $P$ is basically the same as Leibniz's definition of equality, i.e. the identity of indiscernibles. This is not the usual definition.
Mar
1
comment Monad = Reflective Subcategory?
Reflective subcategories are always full in my definition.
Mar
1
comment Monad = Reflective Subcategory?
A monad determines a reflective subcategory if and only if it is idempotent. That is well-known.
Mar
1
comment When to learn category theory?
@Asaf: I don't really like hearing that from you. I don't go around saying set theory is not worth learning, not even as a joke.
Feb
28
comment Analysis and categories
Surely you should at least mention that the standard notion of "continuous functor" is one that preserves limits (in the sense of ordinary category theory)? Or perhaps that there are also internal categories in $\textbf{Top}$ as well as categories enriched in $\textbf{Top}$?
Feb
28
comment Affine algebra of an algebraic group
Actually, an affine algebraic group is a representable functor from the category of $k$-algebras to the category of sets that factors through the forgetful functor $\textbf{Grp} \to \textbf{Set}$.
Feb
28
comment Gentzen Cut elimination: Why do we have to “go infinite”?
There is a subtle difference between the two, though: with the infinitary rule, you can deduce $\forall n . P(n)$ as soon as you have proofs for $P (n)$ for all standard natural numbers $n$. So for example, if you have proofs of sufficiently high complexity (probably $> \epsilon_0$) you could deduce in the infinitary system that $$\forall n . n \text{ does not code the proof of an inconsistency in PA}$$ or something like that.