8
votes
Accepted
Does the forgetful functor $F:\mathrm{CAlg}\to\mathrm{Alg}^{(1)}$ sending $E_\infty$-ring spectra to $E_1$-ring spectra preserve limits and colimits?
Limit-preservation is the content of section 3.2.2 of Higher Algebra (see particularly Corollary 3.2.2.5). Preservation of sifted colimits is in section 3.2.3 (see particularly Corollary 3.2.3.2).
As ...
- 51.5k
7
votes
Accepted
How do these definitions of factorization algebra compare?
I believe that Definition 2 and Definition 3 are equivalent. This involves that Definition 2 implies that F is multiplicative ("for each pairwise disjoint open sets
$U_{1},\dots,U_{k}\subset M$, ...
- 890
6
votes
Accepted
Morita equivalence and connectivity
Suppose $A$ is any nontrivial connective ring spectrum, with right module $P = A \oplus A[1]$, and let $B = End_A(P)$.
The ring $B$ satisfies
$$B \simeq A \oplus A[1] \oplus A[-1] \oplus A$$
and in ...
- 51.5k
5
votes
Fibre sequence of module spectra induces a fibre sequence of $K$-theory spectra?
No, this is not true, and nothing like this is expected.
The exact sequence $\Bbb Z/2 \to \Bbb Z/4 \to \Bbb Z/2$ gives rise to a fiber sequence $H\Bbb Z/2 \to H\Bbb Z/4 \to H\Bbb Z/2$ of Eilenberg--...
- 51.5k
4
votes
Accepted
Is the free algebra functor over an $\infty$-operad symmetric monoidal?
The answer is no in general, and I can't think of reasonable conditions under which it is yes. David's answer was about the "wrong" monoidal structure but it is still helpful : the free ...
- 13.6k
4
votes
Does $\infty$-categorical localization commute with taking directed fibered products?
Here is a counter example in the general case:
Take $E = [1] = 0 \to 1 $. $D = \{0\} \coprod \{1\} $ and $C = \{1\}$. where all maps are weak equivalence.
The lax-pullback is $\{id:1 \to 1\}$, and the ...
- 40.2k
3
votes
Accepted
Diagrams in $(\infty,n)$-categories
I think the answer I want is given by Johnson-Freyd and Scheimbauer's paper "(Op)lax natural transformations, twisted quantum field theories, and 'even higher' Morita categories". Here is a ...
- 1,089
3
votes
How do these definitions of factorization algebra compare?
OK, here is the full story, which confirms what is written in Daniel Bruegmann's answer. In what follows, I will work with a fixed
prefactorization algebra $F$ and an $n$-manifold $M$. I will make
use ...
- 1,803
3
votes
Is the free algebra functor over an $\infty$-operad symmetric monoidal?
It is instructive to think about the simplest possible case, and do away with all the language of $\infty$-operads. If $R$ and $S$ are rings, and we have a ring homomorphism $f:R\to S$, then the left ...
- 29.8k
2
votes
Accepted
Is the symmetric monoidal product on the $\infty$-category of $R$-modules unique?
If by "$R$ is the unit object" you mean : 1- the object $R$ in $Mod_R$ is the unit and 2- The induced commutative algebra structure on $R = map(1,1)$ is the given commutative algebra ...
- 13.6k
2
votes
Accepted
Final and strongly final objects in Higher Topos Theory
It's good to ask this kind of questions on a critical reading! (They are also great exercises in unwinding the definitions to learn to work with simplicial sets, although in the first case I couldn't ...
- 27.2k
2
votes
Quasicategorical Construction of a Cosimplicial Map of Rognes
This question has been answered by the PhD thesis of Aras Ergus. See Corollary 3.2.8 here: https://infoscience.epfl.ch/record/295824/files/EPFL_TH9067.pdf.
The basic idea is to recognize that the ...
- 10.1k
2
votes
Accepted
Different notions of equivalences of $\mathcal{O}$-monoidal $\infty$-categories
They are indeed the same, and in fact you can make the weak version even weaker : if $F$ is strong $O$-monoidal, it suffices to assume that that $F_o$ is an equivalence for every $o\in O^\otimes_{\...
- 13.6k
1
vote
Finitely presentable objects in the categories of algebras of $\infty$-algebraic theories
No, this fails even for the infinity category of spaces, where "free" means "discrete". A coequalizer of free spaces is a wedge of circles, so for example $S^2$ is not of this form....
- 61.5k
1
vote
Accepted
Proof in Higher Algebra that $\mathcal{C}at(\mathcal{K})$ is presentable
Note that by HTTÂ 5.4.1.2 since $\tau$ is an uncountable regular cardinal, an $\infty$-category is $\tau$-compact iff it is $\tau$-small. Our first step is to show that the inclusion $\mathcal{C}at(\...
- 884
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