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bio website math.jhu.edu/~beardsle
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age 28
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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.