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Jun
6
revised Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ preserve (homotopy)colimits?
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Jun
6
comment Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ preserve (homotopy)colimits?
It does matter. $\mathbf{Top}(\Delta^0, -)$ preserves all limits and colimits, for example.
Jun
6
comment Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ preserve (homotopy)colimits?
It's not that trivial: for one thing, the test category is not even a full subcategory of $\mathbf{Top}$.
Jun
6
answered Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ preserve (homotopy)colimits?
Jun
6
comment Category-theoretic cross product and set-theoretic cross product
The categorical product is associative only up to canonical isomorphism. But so is the set-theoretic cartesian product. This causes no problems in practice.
Jun
6
asked Quasicoherent ideal sheaves on open subschemes
Jun
6
comment What (filtered) (homotopy) (co) limits does $\pi_0:\mathbf{sSets}\to\mathbf{Sets}$ preserve?
That's an exercise for you! As a hint, I'll mention that it is a fully faithful functor $\mathbf{Set} \to \mathbf{sSet}$.
Jun
6
revised What (filtered) (homotopy) (co) limits does $\pi_0:\mathbf{sSets}\to\mathbf{Sets}$ preserve?
edited tags
Jun
6
answered What (filtered) (homotopy) (co) limits does $\pi_0:\mathbf{sSets}\to\mathbf{Sets}$ preserve?
Jun
6
comment How to “validate” a Rubik's Cube configuration?
Well, if you have physical access to the cube then you can solve it. There are algorithms that are guaranteed to solve a cube if it is solvable, and if it is not solvable you will reduce it to a state which is manifestly not solvable.
Jun
5
answered Mistake in Popescu's book “Abelian Categories with Applications to Rings and Modules”
Jun
5
comment Does 'let x be a member of S…' require axiom of choice?
@AsafKaragila I presume you have no quibbles with "Let $x \in \emptyset$. [...]", however.
Jun
5
answered Does 'let x be a member of S…' require axiom of choice?
Jun
5
answered horn of a simplex
Jun
4
comment Definition of Zariski Topology
If one is discussing both schemes and classical varieties in the same text, it is probably better to write $\mathbb{A}^n (k)$ for the space of $n$-tuples of elements of $k$ and reserve $\mathbb{A}^n_k$ for the scheme $\operatorname{Spec} k[x_1, \ldots, x_n]$.
Jun
4
comment do you need a Noether ring for Noetherian Theorem?
There is no relation, other than being invented by the same woman.
Jun
4
revised What mathematical objects permit “taking of limits”?
edited tags
Jun
2
comment Can a free group over a set be constructed this way (without equivalence classes of words)?
If you construct the free group using the left adjoint of the forgetful functor $\mathbf{Grp} \to \mathbf{Set}$, you will also not need to mention any equivalence classes of words or reduced words...
Jun
1
answered Is there a more elementary proof of this special case of Riemann-Roch?
Jun
1
comment Is there a more elementary proof of this special case of Riemann-Roch?
Of course, it remains to be shown that elliptic curves are not rational curves. This is usually done using a genus argument, but perhaps the OP does not want to use that...