bio | website | math.jhu.edu/~beardsle |
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location | Baltimore, MD | |
age | 28 | |
visits | member for | 4 years, 6 months |
seen | yesterday | |
stats | profile views | 2,729 |
Learning.
Jun 22 |
accepted | $\Omega X$-action on spectral $X$-bundles |
Jun 22 |
comment |
Lurie's Endomorphism Space vs. Endomorphisms
But yeah, I'm not being complicated at all with the algebra structure on the internal hom object, I'm truly just working through your first paragraph. I guess the point is that I should do it all with universal properties, in this case the universal property of $X^X$. |
Jun 22 |
accepted | Lurie's Endomorphism Space vs. Endomorphisms |
Jun 22 |
comment |
Lurie's Endomorphism Space vs. Endomorphisms
In other words, an arbitrary map $M\to M$ is not necessarily an algebra action of $1_C$ on $M$ (obviously). So, ok. I guess that settles it. |
Jun 22 |
comment |
Lurie's Endomorphism Space vs. Endomorphisms
I think maybe I don't really understand the nature of $C[X]$. I guess I was thinking the objects of $C[X]$ were the data $(C, C\otimes X\to X)$ AND a bunch of higher coherence data. But I guess that that higher coherence data is only put in when you consider algebras of this category. |
Jun 22 |
revised |
Lurie's Endomorphism Space vs. Endomorphisms
added 31 characters in body |
Jun 22 |
comment |
Lurie's Endomorphism Space vs. Endomorphisms
In fact, since an $A$-module is given by a map $N(\Delta)^{op}\times \Delta^1\to Top$, I suspect that the relevant map of algebras is precisely the adjoint $N(\Delta)^{op}\to Map(\Delta^1,Top)$ which picks out the map from $A$ to $End(M)$ as algebras. |
Jun 22 |
revised |
Lurie's Endomorphism Space vs. Endomorphisms
added 102 characters in body |
Jun 22 |
comment |
Lurie's Endomorphism Space vs. Endomorphisms
It may also be fruitful to think about the map $A\times M \to M$ as being part of a simplicial object that Lurie refers to as a "left action object." I'm trying to fiddle with that right now. |
Jun 21 |
comment |
Lurie's Endomorphism Space vs. Endomorphisms
@TheoJohnson-Freyd yes you definitely also need that data. I'm not really mentioning it. I mean, quasicategorically you need a LOT of data (all the higher associativity morphisms), so it'd be nice to get that somehow without specifying an infinite list of $n$-morphisms |
Jun 21 |
revised |
Lurie's Endomorphism Space vs. Endomorphisms
added 266 characters in body |
Jun 21 |
asked | Lurie's Endomorphism Space vs. Endomorphisms |
Jun 17 |
revised |
$\Omega X$-action on spectral $X$-bundles
added 21 characters in body |
Jun 16 |
comment |
$\Omega X$-action on spectral $X$-bundles
@QiaochuYuan that comment is not helpful at all. Obviously I understand that $X\simeq B\Omega X$. If I knew why that implied what I'm describing above, I wouldn't have asked the question. |
Jun 14 |
asked | $\Omega X$-action on spectral $X$-bundles |
Jun 2 |
comment |
Construction of Highly Structured Quotient Groups in Quasicategories
And so one can ask about the universal $E_n$-algebra (or $E_m$-algebra for any $m\leq n$) $Y/X$ such that any map out of $Y$ that lands on an $E_m$-algebra on which $X$ acts trivially factors through $Y/X$. |
Jun 2 |
comment |
Construction of Highly Structured Quotient Groups in Quasicategories
Of course there's an $E_n$-structure to start with, the $E_n$-structure on $Y$ (and on $X$, and on the morphism $X\to Y$). This makes $Y$ into an $E_n$-$X$-algebra in $Top$. |
Jun 1 |
reviewed | Approve Conservativity of multiplicative linear logic over intuitionistic multiplicative linear logic |
Jun 1 |
comment |
Construction of Highly Structured Quotient Groups in Quasicategories
@QiaochuYuan in general an endomorphism space is only a monoid but is there no analogy to be made with the construction of GL_1(R) being an E_n-space when R is an E_n-ring spectrum? |
Jun 1 |
comment |
Construction of Highly Structured Quotient Groups in Quasicategories
@QiaochuYuan I completely changed the question in an attempt to make it a little clearer what precisely I'm looking for. |