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2h
awarded  Notable Question
3h
comment Functors, Objects, Categories
Yes, a functor also has to do things to morphisms; otherwise it would just be a function between sets of objects.
15h
comment Is $(X_G, d_G)$ a compact manifold?
Fair enough. Take $G = S^1$ acting on $X = S^2$ by rotation around a fixed axis.
17h
comment Definition of adjoint functor similar to the definition of homotopy equivalence?
Think of the adjoint functors as continuous functions and the unit and the counit as homotopies.
19h
comment Definition of $K$-linear
Do you know what a linear combination is? An $\mathbb{R}$-linear combination is a linear combination with coefficients in $\mathbb{R}$.
23h
comment Why is this not an inconsistency in elementary Lie theory?
It's surprisingly annoying to write out a full answer, but basically the point is that the $f$ in my first comment is exactly analogous to the $f$ in your second equation. The relationship between how flows of vector fields act on points and how they act on functions is contravariant.
1d
comment $[K(\pi, n), K(\rho, n)] \cong \text{Hom}(\pi, \rho)$
@Michael: as I said, it means $K(A, n)$. This notation emphasizes that $K(A, n)$ is the $n$-fold delooping of $A$ (see, for example, ncatlab.org/nlab/show/delooping and en.wikipedia.org/wiki/Classifying_space).
1d
answered $[K(\pi, n), K(\rho, n)] \cong \text{Hom}(\pi, \rho)$
1d
awarded  Enlightened
2d
comment Lie Group Structure on the 2-Sphere: does the following argument hold?
@Giorgio: ah, you're right, my apologies. It's only solvable. Here is a proof for spheres: math.stackexchange.com/questions/12453/…
2d
revised Lie Group Structure on the 2-Sphere: does the following argument hold?
added 16 characters in body
2d
comment How to calculate homotopy invariant winding number?
You can, for example, find a volume form on $S^2$ with integral $1$, and integrate its pullback along $f$. In general homotopy groups, or for that matter homology groups, contain torsion, so there's no hope for an integral formula. For degrees you can again use volume forms.
2d
answered Lie Group Structure on the 2-Sphere: does the following argument hold?
Aug
31
comment raising elements of profinite groups to $p$-adic powers
@oxeimon: yes, that's correct.
Aug
30
comment How to recover multiplication of group elements from category of groups?
@Zhen: sure, but I'm trying to address the motivating question. My claim is that the Lawvere theory of groups already constitutes "everything about groups." I agree that there's still an interesting additional question remaining.
Aug
30
answered How to recover multiplication of group elements from category of groups?
Aug
30
comment Why is this not an inconsistency in elementary Lie theory?
It's for the same reason that in order to translate the graph of $y = f(x)$ $c$ units to the right, you have to subtract $c$ to get $y = f(x - c)$.
Aug
30
comment Intersection preserves homotopy equivalence
I think the first $X$ was supposed to be a different letter.
Aug
30
answered Intersection preserves homotopy equivalence
Aug
30
comment Nth Homotopy Group Isomorphic to [T^n, X]
$S^n$ and $T^n$ are not even homotopy equivalent (except when $s = 1$; they can be distinguished by $H_1$), so there's no hope for them to be equivalent as anything more structured.