All Questions
3,701 questions
4
votes
0
answers
77
views
Transferring $A_\infty$-structure from a module to its homology
Given an $A_\infty$-module $M$, which is a graded module $M=\bigoplus_{k\in\mathbb{Z}}M_k$ with morphisms $m_n^M\colon A^{\otimes(n-1)}\otimes M\rightarrow M$ of degree 2-n satisfying the $A_\infty$ ...
- 213
0
votes
1
answer
65
views
Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 631
4
votes
0
answers
173
views
Cell structure on the function space $\operatorname{Hom}(X,Y)$
By the Theorem of Milnor in his paper "On spaces having the homotopy type of a CW-complex", the function space $\operatorname{Hom}(X,Y)$ (with the compact-open topology) is homotopy ...
- 12.9k
24
votes
6
answers
2k
views
Multiplicative Structures on Moore Spectra
The motivation for this question is that I want "toy examples" of how to prove/disprove the existence of multiplicative structures on examples of spectra. The class of examples I am thinking of is the ...
- 11.2k
2
votes
0
answers
113
views
Induced homology map zero implies zero in cobordism?
I had asked this in math stackexchange, but got no reply. Hence, I'm asking here.
[I'm no expert in (co)bordism theory, and I've been struggling with it for the past few weeks. Any good references on ...
- 21
8
votes
1
answer
399
views
Reduction of structure group and classifying spaces
Let $H, G$ be topological groups and $\phi : H \to G$ a group homomorphism. Let $M$ be a paracompact topological space.
For any principal $G$-bundle $P \to M$, a reduction (or sometimes 'lift') of its ...
- 95
3
votes
1
answer
246
views
Model structure on simply-connected topological spaces in which the weak equivalences are the rational homotopy equivalences
I recently started learning rational homotopy theory, and found the claim on page 7 of this survey that there is a suitable model category structure in which the weak equivalences are the rational ...
- 37.8k
11
votes
0
answers
427
views
Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
- 631
6
votes
0
answers
211
views
Tate cohomology as a derived functor
I am trying to understand in what sense Tate cohomology may be regarded as the co-fibre of the norm map, from the perspective of derived functors.
Say $G$ is a finite group and $M$ is a $G$-module ...
- 2,907
6
votes
0
answers
147
views
Are cofibrant objects flat with respect to Day convolution?
Question
Let $\mathcal{C}$ be a small symmetric monoidal category. The category $\mathsf{sSet}^{\mathcal{C}}$ of simplicial precosheaves on $\mathcal{C}$ is a symmetric monoidal model category with ...
- 3,901
4
votes
0
answers
101
views
Full subcategories of stable $\infty$-categories closed under all shifts
Is there a name for an $\infty$-category C that admits a zero object and suspensions such that the suspension functor $C \to C$ is an equivalence but which does not necessarily admit cofibers and ...
- 1,361
8
votes
1
answer
437
views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 716
4
votes
0
answers
116
views
Categories enriched in $(m,n)\text{-Cat}$ with Crans–Gray tensor product
(All higher categories will be strict unless otherwise noted):
It is a somewhat folklorish fact that if you equip the category of $(m,n)$ categories with the cartesian monoidal structure, then a ...
- 12.9k
3
votes
0
answers
170
views
Cellular structure of $F_4$
Is there the cellular structure of the Exceptional Lie group $F_4$?
Is there a reference to it?
Thanks
3
votes
2
answers
374
views
Is this true of the frame bundle $\operatorname{Fr}(M)$?
On an orientable (Riemannian) $n$-manifold $M$, with orthonormal frame bundle $\operatorname{Fr}(M)$, we have that the tangent bundle classifying map $\tau_M : M \to B{\operatorname O(n)}$ lifts to $B{...
- 95
0
votes
0
answers
80
views
Reference for computing cohomology of $Q_nS^0$ (the $n$-th component of infinite loop space)?
$Q_nS^0=\varinjlim_k(\Omega^kS^k)_n$, where $(\Omega^kS^k)_n$ refers to homotopy classes of maps $(S^k,\ast)\to (S^k,\ast)$ of degree $n$. I already know the case of $n=0$.
Is there any reference ...
- 869
4
votes
2
answers
458
views
Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces
Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
13
votes
2
answers
391
views
What does $\mathrm{Conf}_n(M)^{h S_n}$ look like?
$\DeclareMathOperator\Conf{Conf}$Let $M$ be a manifold, and $\Conf_n M$ the ordered configuration space of n points on $M$. The symmetric group $S_n$ acts by permuting the points.
Is there a simple ...
- 56.9k
5
votes
2
answers
178
views
Quasi-equivalent vs. homotopy equivalent functors in $A_\infty$ categories
Suppose that $\mathcal{A}, \mathcal{B}$ are strictly unital $A_\infty$ categories, and $\mathcal{F}, \mathcal{G}: \mathcal{A} \rightarrow \mathcal{B}$ are (strictly unital) functors.
On one hand, we ...
- 196
3
votes
1
answer
234
views
Do the two orientations on an orientable manifold $M$ witness the same lifts of $\tau_M: M \to B\text{O($n$)}$ to $B\text{SO($n$)}$?
For an orientable $n$-manifold $M$ and its (orthonormal) frame bundle classifying map $\tau_M : M \to BO(n)$, we have a lift diagram of the following sort:
There are two orientations on $M$. Is it ...
- 18.8k
4
votes
1
answer
237
views
When does a cofibrantly generated model category have this factorization property?
Let $\mathcal{C}$ be a cofibrantly generated model category, which is generated by $I$ and $J$. According to the small object argument (Hovey Theorem 2.1.14) of cofibrantly generated model categories, ...
- 42.4k
2
votes
0
answers
92
views
Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
4
votes
1
answer
364
views
Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$
Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread.
Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...
CommunityBot
- 1
0
votes
0
answers
61
views
Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO.
I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
- 63.9k
4
votes
1
answer
375
views
Is the geometric realization of simplicial functors interesting?
While studying a completely unrelated problem, I have proved something on the following line: given a diagram of simplicial sets $X: C \to \textrm{sSet}$, some deformations of the geometric ...
- 2,152
5
votes
0
answers
201
views
Replacing $\mathbf{Cat}$ with a $2$-category in Grothendieck construction
Let $\mathbf{Cat}$ denote the (strict) $2$-category of categories, functors, and natural transformations. Let $F: C \to \mathbf{Cat}$ be a pseudofunctor from a $1$-category $C$ to $\mathbf{Cat}$. The ...
- 10.7k
8
votes
1
answer
470
views
Finite domination and compact ENRs
Edit: In the comments, Tyrone points out that West's positive answer to Borsuk's conjecture implies that every compact ENR is homotopy equivalent to a finite CW complex. It follows that the only ...
- 5,596
3
votes
0
answers
145
views
What is the group completion of the underlying multiplicative $\mathbb{E}_\infty$-monoid of the sphere spectrum?
I recently noticed the following categorical/universal way to describe the passage from $\mathbb{Z}$ to $\mathbb{Q}$:
We start with the categroy $\mathsf{Sets}^{\mathrm{actv}}_*$ of pointed sets and —...
- 11.8k
19
votes
1
answer
1k
views
Steenrod squares as power operations vs. as cohomology operations
There are already several excellent questions and answers on MO regarding Steenrod squares, understanding them in various ways and relating them to power operations and I think I get this. Still, I am ...
- 18.4k
3
votes
1
answer
184
views
Freudenthal suspension homomorphism
I asked this question in MathStackExchange a couple of months ago, with little feedback, hence I try here.
The Hopf invariant $h(f)$ of a mapping $f:\mathbb S^{2m-1}\to \mathbb S^m$ is a homotopy ...
- 203
9
votes
1
answer
425
views
Delta-generated spaces vs CW complexes
$\newcommand\Top{\mathrm{Top}}\newcommand\CW{\mathrm{CW}}\newcommand\Deltagenerated{\text{$\Delta$-generated}}\newcommand\Spaces{\mathrm{Spaces}}\newcommand\DeltaSpaces{\text{$\Delta$-Spaces}}$I am ...
- 7,193
6
votes
2
answers
358
views
Transfer model structures, reflective subcategories and Bousfield localizations
The topic of Bousfield localizations has a lot of literature which has on most of the occasion some tameness assumption on the presentability of the model category. Recently I have been trying to ...
- 40.2k
4
votes
1
answer
239
views
True or false? Every left or right cancellative, duo semigroup is cancellative
A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
- 10.5k
2
votes
1
answer
404
views
Reference request: a cousin to the log semiring
Let $f$ be strictly increasing on $\mathbb{R}$. Then $x \oplus y := f^{-1}(f(x)+f(y))$ gives rise to a strict symmetric monoidal ($\Rightarrow$ commutative monoid) structure on $(\mathbb{R},\ge)$ with ...
CommunityBot
- 1
6
votes
0
answers
128
views
Induced map of degree $k$ self map of a sphere in the higher homotopy groups
Let $f:S^n\rightarrow S^n$ be a degree $k$ map. Then $f$ induces maps $\pi_l(S^n)\rightarrow \pi_l(S^n)$.
I believe that in the stable range ($l\leq 2n-2$) this map is multiplication by $k$. Unstably ...
- 7,583
7
votes
0
answers
266
views
Connecting Harris's proof of periodicity to the Bott element
$\newcommand\ku{\mathrm{ku}}$In Denis Nardin's lectures on stable homotopy theory, he reproduces Bruno Harris's proof of Bott periodicity from group completion and goes on to identify the Bott element ...
- 1,337
2
votes
0
answers
114
views
The induced map between push outs in an exact infinity category
Let $(\mathcal{C} , \mathcal{M} , \mathcal{E})$ be an exact $\infty$-category. (I am following the definition in Higher Segal Spaces $I$ by Dyckerhoff and Kapranov). Assume that $F$ is a cofiberation ...
- 167
4
votes
1
answer
469
views
How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?
$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes:
https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf
https://...
- 2,806
1
vote
0
answers
97
views
Postnikov invariant of crossed square
Is there a reference where Postnikov invariants of the classifying space of a crossed square have been computed ? I am especially interested in the computation of the third Postnikov invariant $B\...
7
votes
1
answer
473
views
Why do homotopy orbits commute with the Eilenberg–Mac Lane spectrum functor?
Let $H: \mathsf{sAb}\to H\mathbb{Z}\text{-}\mathsf{Mod}$ denote the Eilenberg–Mac Lane functor sending a simplicial abelian group $M_\bullet$ to the infinite delooping of its geometric realization, i....
- 56
4
votes
1
answer
318
views
Functoriality of infinite suspension spectrum functor on infinity groupoids!
Consider the functor $F: C \rightarrow D $ of $\infty$-groupoids. Is there any explicit proof somewhere in the literature that $\Sigma^{\infty}$ construction is functorial? I mean how do we define $\...
- 1,337
6
votes
3
answers
551
views
Conjecture about commutative semigroups
Conjecture: given any commutative semigroup $S$ of order $n \ge 4$, there exist $a, b \in S$ with $a \ne b$, an integer $m \ge \lfloor (n-1)/2 \rfloor$, and two $m$-element subsets $X = \{x_1, \ldots, ...
- 38.6k
6
votes
0
answers
318
views
Animating cocartesian fibrations
Let $F: \mathcal{E}\to \mathcal{C}$ be a cocartesian fibration of ordinary categories satisfying certain conditions so that we can take the animation of $F$. How to show that $\mathrm{Ani}(F)$ is a ...
- 711
5
votes
1
answer
197
views
Examples of cyclic A-infinity algebra
I am wondering about (references to) examples of cyclic A-infinity algebras- especially including explicit descriptions of the structure maps and pairing.
Thanks a lot!
- 121
4
votes
0
answers
125
views
Minimal model for $A_\infty$-categories
Is there a reference for existence and construction of the minimal model of an $A_\infty$-category? Most references I found ultimately refer to Lefèvre-Hasegawa's thesis but there doesn't seem to be a ...
- 489
8
votes
1
answer
322
views
Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
7
votes
2
answers
488
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
8
votes
2
answers
596
views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 38.6k
3
votes
0
answers
250
views
Action (of a graded monoid) required
Reference request: Did the construction below appear anywhere before? Any mentions of it or especially any links to something commonly known would be really helpful. I feel that it might be related to ...
- 321
2
votes
0
answers
169
views
Formal power series over the sphere spectrum?
In Section 11 of their paper https://arxiv.org/pdf/1802.03261, Bhatt-Morrow-Scholze discuss the polynomial algebra over the sphere spectrum. I'm wondering whether its possible to define a notion of ...
- 959