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28m
comment What categorical limits and colimits does $\pi_1$ preserve?
$\pi_1$ is representable in the homotopy category of pointed spaces, so it preserves all limits in that category. Unfortunately, there aren't so many of those. I would not be so quick to say that it preserves coproducts either; after all, van Kampen's theorem only applies under certain niceness hypotheses.
2h
comment A contravariant functor taking colimits to limits is representable.
No, you do not need the adjoint functor theorem. Just go look at some examples first!
12h
comment A contravariant functor taking colimits to limits is representable.
Well, start with a representable functor $\mathbf{Set}^\mathrm{op} \to \mathbf{Set}$. Can you figure out how to recover the representing object from just the functor? Then apply this procedure to your contravariant functor.
1d
comment Functoriality of fundamental group
It suffices to be able to choose skeletons for connected groupoids functorially. However it is not obvious to me whether this is possible or not; it may even be independent of set theory...
1d
comment How many isomorphisms from a set to itself
Never mind that, $\left| \mathrm{Hom} (X, X) \right|$ is not $n^2$ but rather $n^n$.
2d
comment What is the opposite category of $Set$?
The opposite category is the opposite category. There is no need to think about morphisms as actual functions with the specified domain and codomain; and indeed, you can't.
Oct
19
comment Infinite Cardinal Addition Without the Axiom of Choice
You can't construct such an example without using some extra set-theoretic principles.
Oct
16
comment Is the inverse limit of simplicial maps between finite directed graphs also a graph?
Hmmm. Let $G_n$ be the graph with vertices $\{ 0, \ldots, n \}$ and one edge connecting $0$ to each $i > 0$. Define maps $f_n : G_{n+1} \to G_n$ by collapsing $n$ and $n + 1$; then the inverse limit has infinitely many vertices. Does this fit your criteria?
Oct
16
comment Is the inverse limit of simplicial maps between finite directed graphs also a graph?
I'm confused. By "topological inverse limit", do you mean inverse limit in $\mathbf{Top}$? In which case, are you identifying graphs with their geometric realisations?
Oct
16
comment Yoneda implies $\text{Hom}(X,Z)\cong \text{Hom(}Y,Z)\Rightarrow X\cong Y$??
It also has to be a natural bijection, or at least, natural enough.
Oct
16
comment Variety of Connected Components
Isn't it clear that $\pi_0 (X)$ should be a disjoint union of $n$ copies of $\operatorname{Spec} k$, where $n$ is the number of connected components?
Oct
15
comment Let $\pi: E \to M$ a vector bundle. Is $E$ a direct summnad of $M\times\mathbb{R}^{d}$, for some $d$?
Note that the base is assumed to be compact Hausdorff.
Oct
15
comment Can a model (of a general theory) be viewed as a (less general) theory?
A group does not "contain" representations. Moreover, representations are not themselves groups but rather homomorphisms of groups. There are many other objections. Please read an introduction to first-order logic.
Oct
15
comment kan extension,(co)ends,natural bijection
Did you look at [CWM, Ch. X §4]?
Oct
15
comment Some Galois theory
Search for "simple extensions".
Oct
14
comment Singular homology of cofinite topology space
My impression is that these spaces are contractible.
Oct
13
comment How can I visualize principal bundles?
@NajibIdrissi It is the easiest non-trivial example of a principal $G$-bundle that can be embedded in $\mathbb{R}^3$.
Oct
13
comment How can I visualize principal bundles?
There is a unique non-trivial principal $O(1)$-bundle over $S^1$, namely the unique connected double cover of $S^1$. This is easy enough to visualise.
Oct
12
comment Confusion in basic defintion of sheaf cohomology
The meaning of exactness depends on the category.
Oct
11
comment (Are there) subtleties in the definition of 'biproduct'
Well, consider an infinite set $X$. Then for cardinality reasons, $X + X \cong X \times X$, yet we do not say that it is a biproduct.