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 Curious
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Jan
13
comment $\mathcal{V}$-naturality in enriched category theory
Thank you! Now I know I'm not the only one :), I thought Max Kelly's book was like the most basic and introductory book on the subject, are there other good introductory books out there that I can use as a reference too?
Jan
13
comment $\mathcal{V}$-naturality in enriched category theory
Sorry for taking so long, had a bit of a family emergency. So that's what he means by "$\mathcal{V}$-natural in each variable", so the functors $(- \otimes Y) \otimes Z$ and $- \otimes (Y \otimes Z)$ with the natural transformation $a_{X,Y,Z}$ define functors $(- \otimes Y) \otimes Z_{enr}$ and $- \otimes (Y \otimes Z)_{enr}$ and a $\mathcal{V}$-natural transformation $a_{enr}$ between them, right? I tried to prove this manually and got the corresponding $\mathcal{V}$-naturality condition in 1.7, but I used (1.28) to get there, not (1.34), so I don't get how it "follows easily from (1.34)"?
Jan
6
comment $\mathcal{V}$-naturality in enriched category theory
If $\mathcal{V}$-naturality refers to a dinatural (extranatural) transformation between two functors, what does he mean by "the map $a: (X \otimes Y) \otimes Z \rightarrow X \otimes ( Y \otimes Z)$ is $\mathcal{V}$-natural in every variable"? Where are the two functors there? All I see is a map $a$ in the category $\mathcal{V}$.
Jan
5
asked $\mathcal{V}$-naturality in enriched category theory
Mar
18
accepted Topological modules and relative homological algebra.
Mar
18
comment Topological modules and relative homological algebra.
It's a lot more clear now, thanks again.
Mar
14
comment Topological modules and relative homological algebra.
That's fine thanks, but back to my original question, the one thing I don't understand is why would Taylor say that his theory is an example of relative homological algebra if it isn't? There's a class of allowable short exact sequences but the categories involved $\mathcal{A}$ and $\mathcal{B}$ are additive, but Mac Lane defines relative homological algebra, relative abelian categories, and allowable short exact sequences when the categories involved are abelian, or I guess it doesn't matter since the techniques are still useful if we want to do homological algebra in this context?
Mar
13
comment Topological modules and relative homological algebra.
Sorry for taking so long to respond, thanks for the answer, so here $\mathcal{A}$ = the category of locally convex $A$-modules = $A$-mod, and $\mathcal{B}$ = the category of $C$-modules (relative to $\widetilde{\otimes}$) = topological vector spaces over $C$ (which are not abelian categories, but they are additive). Honestly, what I wanted to know was if the category of chain complexes in $A$-mod has the structure of a model category.
Mar
8
revised Topological modules and relative homological algebra.
edited title
Mar
8
asked Topological modules and relative homological algebra.
Mar
8
accepted What is the coproduct in the category of Banach spaces and continuous linear maps?
Jul
2
awarded  Curious
Jun
27
comment What is the coproduct in the category of Banach spaces and continuous linear maps?
Thank you, will see if I can find the question.
Jun
27
comment What is the coproduct in the category of Banach spaces and continuous linear maps?
Thank you both, great insight, I'll look at your blog entry to get up to speed.
Jun
27
asked What is the coproduct in the category of Banach spaces and continuous linear maps?
Mar
11
accepted Strong Morita equivalence - Question about proof in Beer's “On Morita equivalence of nuclear $C^*$-algebras”
Mar
6
comment Strong Morita equivalence - Question about proof in Beer's “On Morita equivalence of nuclear $C^*$-algebras”
Thank you, I suppose I can find that (and more on the subject) in an introductory level book on Hilbert modules? Thanks for your answer.
Mar
1
asked Strong Morita equivalence - Question about proof in Beer's “On Morita equivalence of nuclear $C^*$-algebras”
Mar
26
awarded  Scholar
Mar
26
accepted Resolutions of bimodules as $R^e$-modules.