All Questions
20 questions
33
votes
0
answers
2k
views
Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?
For a (discrete) monoid $M$, the classifying space $BM$ is the
geometric realization of the nerve of the one object category whose
hom-set is $M$. (This definition gives the usual classfiying space
...
- 5,309
16
votes
2
answers
1k
views
The symmetric monoidal category of finite sets
It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{FinSet}$ of finite ...
- 63.1k
13
votes
2
answers
713
views
How do you compute the space of lifts of an E-infinity map?
Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps $g:...
- 3,103
13
votes
0
answers
251
views
Is every simply connected finite complex the classifying space of a finite monoid
On page 323 of Fiedorowicz, "Classifying Spaces of Topological Monoids and Categories" it was stated that "it seems likely that any finite simply connected complex should [have the same weak homotopy ...
- 665
10
votes
2
answers
1k
views
Connective spectra versus simplicial abelian groups - very basic question
Hello,
I have very general , "introductory" questions (It is quite hard for me to seek for specific things in the algebraic topology literature).
I guess that connective spectra have a model ...
- 5,562
10
votes
1
answer
274
views
A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids
In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following.
...
- 38.6k
10
votes
1
answer
440
views
Reference for a generalization of Γ-spaces to monoidal model categories
Γ-spaces were introduced by Segal in 1969 as models for what can be now described
as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...
- 37.8k
7
votes
1
answer
593
views
Question about topological monoid maps
Let Mon be the category of topological monoids. I am happy to work with the model structure mentioned here:
Model Structure/Homotopy Pushouts in topological monoids?.
I'm looking for a reference ...
- 18.8k
6
votes
1
answer
444
views
Homotopy type of a specific discrete monoid
Consider the discrete monoid $M$ of nondecreasing continuous maps from $[0,1]$ to itself preserving the extremities. Note that the monoid is right-cancellative ($x.z=y.z$ implies $x=y$, since $z$ is ...
- 3,901
5
votes
1
answer
170
views
Which homotopy types can be realized as the classifying space of a right-cancellative discrete monoid?
McDuff showed that every connected homotopy type can be realized as the classifying space of a discrete monoid, but the monoid she constructs has lots of idempotents.
Question: Which homotopy types ...
- 63.9k
5
votes
1
answer
284
views
Directed homotopy in the Cayley graph of a monoid
There is a the notion of the Cayley graph $C(G)$ of a group $G$ (which depends on a given presentation $G \cong \mathcal F(S) / \sigma$ where $\mathcal F$ is the free group functor and $\sigma$ some ...
- 190
5
votes
1
answer
205
views
Topological category of topological monoids / operads
The category of topological monoids can be made into a topological category in a naive way. Namely, the space of all continuous homomorphisms between two topological monoids is a closed subspace of ...
- 53
5
votes
2
answers
252
views
Monoid of continuous self-maps of (real) surfaces
Let $S$ be a closed surface of genus $g > 0$ and $[S,S] = Hom(\pi_{1}(S),\pi_{1}(S))$ be the monoid of (homotopy classes of) continuous maps from $S$ to itself. Consider the semi-group $A$ of ...
- 6,995
5
votes
1
answer
304
views
flat maps of monoids which are not localizations
It is well known that a localization $S^{-1}R$ of a commutative ring $R$ is flat as a $R$-module.
Rather, I am looking for extensions of rings which share certain properties of localizations, like ...
- 6,210
4
votes
1
answer
246
views
group completion theorem of homology as Hopf algebras
Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a ...
- 2,223
3
votes
1
answer
161
views
Are cofibrations in topological monoids preserved by forming the product with the identity?
Consider the category $\mathrm{Mon}(\mathbf{Top})$ of topological monoids, together with the model structure transferred along the adjunction $F:\mathbf{Top}\rightleftarrows \mathrm{Mon}(\mathbf{Top}):...
- 1,623
3
votes
0
answers
176
views
The monoid of stably-free modules over integral group rings
Fix a torsion-free group G, let $M_G$ be the monoid of stably-free $\mathbb{Z}G$-modules under operation $\oplus$, the direct sum of modules.
In studying objects related to Wall’s D2 problem on CW-...
- 187
3
votes
0
answers
163
views
Classifying spaces of amalgamated topological monoids
Let $\mathsf{Top}_*$ be the category of well-based spaces and $\mathsf{TopMon}$ the category of topological monoids. Recall the James construction $\mathcal{J}:\mathsf{Top}_*\to \mathsf{TopMon}$ which ...
- 1,623
2
votes
0
answers
180
views
Pro-p topology on free group
Let $H$ be a finitely generated subgroup of the free group $F(A)$ and $G_P$ the pseudovariety of all finite $p$-group with $p$ fixed prime number. We endow $F(A)$ with the pro-$G_p$ topology. Suppose ...
- 421
1
vote
0
answers
139
views
Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
- 12.5k