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Part of higher category theory that for instance in Algebraic Topology enables us to capture finer homotopic distinctions. As in say Eilenberg-Maclane spaces.
4
votes
1
answer
316
views
Bar Construction Model of Ring Spectrum Quotient
Suppose I am given a morphism $f:BG\to BGL_1(R)$ for $R$ some at least $E_1$-ring spectrum and $G$ a loop space. Then This corresponds, I believe, to an action of $G$ on $R$, coming from a morphism $G …
14
votes
2
answers
960
views
Are n-truncated quasicategories a model for n-categories?
In Higher Topos Theory, Lurie gives a description of $n$-categories (section 2.3.4) in terms of quasicategories. In other words, he gives a definition (2.3.4.1) of an $\infty$-categorical notion of $n …
10
votes
0
answers
329
views
When is Fun(X,C) comonadic over C with respect to the colimit functor?
Because I'm primarily interested in this question from the point of view of $\infty$-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular …
11
votes
1
answer
833
views
What does the homotopy coherent nerve do to spaces of enriched functors?
Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a simp …
3
votes
0
answers
258
views
Right Notion of Localizing Subcategory in Quasicategory
Given a stable quasicategory $C$, what are the conditions required of a subcategory for it to descend to a localizing subcategory in $Ho(C)$? Is it enough for it to be reflective? Perhaps exact and re …
2
votes
1
answer
248
views
Showing left module actions are highly structured
For my own convenience I'll work in $\infty$-categories, feel free to answer in whatever framework best suits you. My question is essentially how to show, given an $E_\infty$-ring object $R$ in an $\i …
4
votes
1
answer
541
views
Straightening for $\infty$-operads
There is this straightening/unstraightening procedure of Jacob Lurie's which takes a symmetric monoidal $\infty$-category (which is the data of a coCartesian morphism of simplicial sets $C^\otimes\to …
2
votes
Accepted
Straightening for $\infty$-operads
This question was answered in the affirmative by Rune Haugseng in Section 4 of https://arxiv.org/pdf/1708.09632.
1
vote
Accepted
Higher Degree Data in a Cosimplicial Quasicategory and Delooping
So, as far as I can tell, this question has a lot of high-falootin' vocabulary in it, but is actually pretty basic. It just took me a while to think about the right way. I should mention that my under …
1
vote
Why does $Mf$ always support an $Mf$-orientation?
I just want to add another answer to this, which is Corollary 4.16 of arXiv:1810:00734. This result of course uses Omar and Tobias' in an essential way, so is not somehow independent, but what it does …
4
votes
Accepted
Lifting Strict Comonoids and Comodules to Quasicategories
The answer to this question is yes, and it's the main result of this paper.
One thing to point out is that, even in the case that the tensor product of $\mathcal{M}$ preserves fibrant objects (so tha …
4
votes
1
answer
274
views
Higher Degree Data in a Cosimplicial Quasicategory and Delooping
If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference.
My question is regarding accessing data i …
12
votes
2
answers
558
views
Why does $Mf$ always support an $Mf$-orientation?
Let $f:X\to BGL_1(\mathbb{S})$ be a morphism of $E_n$-spaces and determine a principle $GL_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom isomor …
8
votes
1
answer
332
views
Lifting Strict Comonoids and Comodules to Quasicategories
$\newcommand{\M}{\mathcal{M}}$
Suppose I have a monoidal simplicial model category in which every object is cofibrant $(\M,\otimes,\mathbb{1})$ and I want to look at its underlying monoidal quasicate …
4
votes
1
answer
948
views
Opposite Symmetric Monoidal Structure on an Infinity Category
Given an $\infty$-category (in the sense of Lurie) $C$, and a symmetric monoidal structure on $C$ associated to a coCartesian fibration $p:C^\otimes\to N(Fin_\ast)$, Lurie says in Remark 2.4.2.7 of Hi …