bio | website | |
---|---|---|
location | ||
age | ||
visits | member for | 2 years, 4 months |
seen | 19 hours ago | |
stats | profile views | 186 |
Mar 22 |
awarded | Informed |
Mar 10 |
comment |
In what generality does Eilenberg-Watts hold?
Thanks for the tip about dense generators, Zhen. That cleared everything up for me. So Eilenberg-Watts lemma is just another statement of the Yoneda lemma, basically? That's neat. |
Mar 8 |
comment |
In what generality does Eilenberg-Watts hold?
@ZhenLin: Yes, the representables are generators of the presheaf category. But I still don't see how to proceed by analogy with the ring module case without some addition lemmas. In an arbitrary category with generator $P$, can we write every object as a colimit of a constant functor at $P$? I guess that would be sufficient to give me what I want, but it seems stronger than the definition of generator I know (either $\hom(P,-)$ is injective on arrows or every object receives an epimorphism from $\coprod P$) |
Mar 8 |
asked | In what generality does Eilenberg-Watts hold? |
Jan 29 |
revised |
comparison between two monadic definitions for an operad
reword title slightly |
Jan 29 |
asked | comparison between two monadic definitions for an operad |
Jan 20 |
comment |
Maximal ideals in the ring of continuous real-valued functions on R
So why doesn't this apply to the maximal ideal containing the compactly supported functions? The residue field is not $\mathbb{R}$? What is it? The field of functions which don't vanish at infinity? |
Jan 12 |
awarded | Popular Question |
Jan 6 |
awarded | Excavator |
Jan 6 |
revised |
Weighted limits and completeness
Mike's link to his Chicago homepage is dead. I update URL to same PDF on his San Diego webpage. |
Jan 6 |
suggested | suggested edit on Weighted limits and completeness |
Dec 26 |
accepted | Correspondence between operads and monads requires tensor distribute over coproduct? |
Dec 26 |
comment |
Correspondence between operads and monads requires tensor distribute over coproduct?
Thank you for clarifying, gentlemen. I had thought requiring monoidal product to preserve coproducts would be an unusual requirement, which is why I had to ask. I somehow forgot that this was implied by closedness of the monoidal product, which is a natural for enrichment categories. So obvious in hindsight... thanks again! |
Dec 25 |
asked | Correspondence between operads and monads requires tensor distribute over coproduct? |
Dec 18 |
comment |
Gaussian curvature and mean curvature.
This is number 17 in section 3.3 on page 172 of do Carmo's book on the differential geometry of curves and surfaces. Verbatim. |
Nov 7 |
comment |
Classification of 1-dimensional manifolds (not second-countable)
Thanks, Todd. I'm on board now. |
Nov 7 |
comment |
Classification of 1-dimensional manifolds (not second-countable)
@ToddTrimble: but how do you know that $S^1$ is the only 1-manifold which satisfies this property, that removing a point leaves one connected component? That seems like the most important part of the classification to me. |
Nov 5 |
comment |
Classification of 1-dimensional manifolds (not second-countable)
@ToddTrimble: you start out with "let $Y$ be a connected nonempty topological 1-manifold. Then $Y-\{y\}$ has two components." I don't understand. $S^1$ is a connected nonempty topological 1-manifold, but $S^1-\{\text{pt}\}$ has one component...? |
Sep 30 |
comment |
Euler Sequence on Homogeneous Spaces
The fiber of $H\otimes \mathcal{O}(1)$ is $V/[x]\otimes [x]^*.$ So let $\sigma\in [x]^*$ and $v+[x]\in V/[x].$ Define the map to be $\sigma\otimes(v+[x])\mapsto \sigma(x)\cdot d\pi_x(v).$ Then show that this is well-defined as a function on $P(V)$, i.e. $\sigma(x)\cdot d\pi_x(v) = \sigma(\lambda x)\cdot d\pi_{\lambda x}(v).$ But is there a more elegant way to say it, i.e. not in terms of an arbitrarily chosen point $x$ in $V$? |
Sep 30 |
comment |
Euler Sequence on Homogeneous Spaces
Can you elaborate about how we derive $T_{P(V)}\simeq H\otimes \mathcal{O}(1)$ from the differential of the projection map? Are we to understand $d\pi_x$ as the linear map on a fiber of a bundle map $H\otimes \mathcal{O}(1)\to T_{P(V)}$? |