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visits member for 2 years, 4 months
seen 19 hours ago

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 $H\otimes \mathcal{O}(1)\to T_{P(V)}$?