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Dec
19
comment Is there a classification of regular maps $\mathbb{P}^1(k)\to\mathbb{A}^1(k)$?
They are all constant maps.
Dec
17
comment A ``partial'' Mitchell-Benabou language?
If you can only classify certain subobjects, then in your semantics, you can only use those subobjects. But I don't really see any problems with that if your class of subobjects is sufficiently nice.
Dec
17
comment The tensor product of monads.
No. Wishing for something doesn't make it true!
Dec
17
comment Subobjects Equivalent iff Isomorphic Domains?
This is false. Try coming up with two non-isomorphic subobjects $1 \to 2$ in $\mathbf{Set}$.
Dec
17
comment The tensor product of monads.
So, you're asking why the cartesian product is not the same as composition? Well, one is commutative and the other isn't...
Dec
17
comment Colimit in the category of (all) simply transitive group actions
Well, products are not a problem, so it must equalisers. And sure enough, if you think about parallel pairs in $\mathcal{C}$ whose equaliser is empty, you will see that they have no equaliser in $\mathcal{C}_r$.
Dec
17
comment Quotient Objects in $\mathsf{Grp}$ II
It means that the isomorphism is compatible with the inclusion (resp. quotient) map.
Dec
17
comment Quotient Objects in $\mathsf{Grp}$ II
If $M$ and $N$ are isomorphic as subgroups then $G / M$ and $G / N$ are isomorphic as quotients.
Dec
16
comment Triangulation of hypercubes into simplices
@Spenser Five is possible. I didn't say the tetrahedra were all the same size and shape.
Dec
16
comment Triangulation of hypercubes into simplices
Where does "6 tetrahedrons" come from? There is a subdivision into 5.
Dec
15
comment Reference for $(\infty,1)$-Categories
The classic reference for enriched category theory is [Basic concepts of enriched category theory]. I do not know of any reference developing algebraic topology from this point of view – but then again, I do not think it would be helpful.
Dec
15
comment Naturality of Transformations
It seems to me that this is just a verbose way of saying "dummy variable".
Dec
15
comment Reference for $(\infty,1)$-Categories
I must strongly recommend learning some basic enriched category theory first. It will be conceptually clearer that way.
Dec
12
comment About certain regular epimorphisms in a Grothendieck Topos
I was also quite surprised at how fiddly it is to write out the proof properly. In my view the difficulty is entirely contained in determining when two elements become equal in the pushout.
Dec
12
comment relation between first fundamental form for different parametrization
The point is that, for coordinates $(u, v)$ and $(u', v')$ and respective f.f.f. $(E, F, G)$ and $(E', F', G')$, we get $E \, \mathrm{d}u^2 + 2 F \, \mathrm{d}u \, \mathrm{d}v + G \, \mathrm{d}v^2 = E' \, \mathrm{d}u'^2 + 2 F' \, \mathrm{d}u' \, \mathrm{d}v' + G' \, \mathrm{d}v'^2$.
Dec
12
comment relation between first fundamental form for different parametrization
Use the chain rule. (It's unfortunate that you've chosen the same letters for both parametrisations.)
Dec
12
comment Contradictory Orientations of Faces in Simplicial Complexes
The only purpose in defining the orientation of simplices is to define the boundary of a chain, and in order to have a chain complex, we need $\partial \circ \partial = 0$. For this to happen, we need a lot of cancellation to happen, so it is a good thing that the signs disagree.
Dec
12
comment Contradictory Orientations of Faces in Simplicial Complexes
Have you not seen the construction of simplicial homology?
Dec
12
comment Module of differentials in the functorial approach to schemes and quasi-coherent modules
Is your formula correct? I would rather guess $\int_A \prod_{x \in X (A)} \mathrm{Der}_R (A, M (x))$, and even then it's not obvious that it's right. Anyway, sometimes things have both a left universal property and a right universal property.
Dec
12
comment Contradictory Orientations of Faces in Simplicial Complexes
Actually, they are supposed to be contradictory. Otherwise you wouldn't get a chain complex.