All Questions
Tagged with monads at.algebraic-topology
12 questions
26
votes
1
answer
2k
views
Infinity-categorical analogue of compact Hausdorff
Recently I became through this mathoverflow question aware of the article Codensity and the ultrafilter monad by Tom Leinster. There he shows that the ultrafilter monad on the category $\mathrm{Set}$ ...
- 12.1k
21
votes
0
answers
1k
views
Can we describe equivariant vector bundles of free group action in terms of descent theory (Barr-Beck theorem)?
It is well known that for a compact topological group $G$ acts (say, from the right) freely on a compact space $X$. Then the category of equivariant complex vector bundles on $X$, $\text{Vect}_G(X)$, ...
- 9,019
11
votes
2
answers
755
views
When does the forgetful functor from algebras over a monad commute with homotopy geometric realizations?
Let $\mathcal{C}$ be a combinatorial model category and $\mathrm{T}$ a monad on
$\mathcal{C}.$
Assume that the model structure on $\mathcal{C}$ lifts to a model structure on
the category of $\...
- 1,361
9
votes
2
answers
739
views
Correspondence between operads and monads requires tensor distribute over coproduct?
In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without ...
- 1,446
9
votes
2
answers
366
views
monadic decomposition
Let $\mathrm{F}: \mathcal{C} \rightleftarrows \mathcal{D} : \mathrm{G} $ be an adjunction with associated monad $\mathrm{T} = \mathrm{G} \mathrm{F} .$
If $\mathcal{D} $ admits coequalizers of $\...
- 1,361
7
votes
2
answers
858
views
What are the algebras over $\Omega^k\Sigma^k$ ?
Let $Ho(Spc)$ be the homotopy category of spaces. There is an adjoint pair
$$
\Sigma^k \colon Ho(Spc) \leftrightarrows Ho(Spc)\colon \Omega^k,
$$
where $\Sigma^k$ is the $k$-th supension functor and $\...
- 101
6
votes
2
answers
848
views
Monadicity theorem in homotopy theory.
Let $\mathbf{C}$ be a cofibrantly generated model category (assume for simplicity that all objects are fibrant) and $\mathbf{C}^{\mathrm{T}}$ the category of $\mathrm{T}$-algebras with the induced ...
- 1,974
5
votes
0
answers
166
views
Algebras of the cone monad on Top?
Let us work in Top, the category of topological spaces - although the reader is welcome to replace this by their favorite convenient category of topological spaces.
If $X,Y$ are spaces, let $X\ast Y$ ...
4
votes
1
answer
763
views
When do reflexive coequalizers preserve weak equivalences?
In my work I've run into the following situation. In a model category, I have two reflexive coequalizers $A_i \stackrel{\to}{\to} B_i \to C_i$ and a map of diagrams which is levelwise a weak ...
- 30.3k
4
votes
1
answer
391
views
Different ways to “deloop” a (topological) $A_\infty$-algebra
Let $\varphi:A\to \mathrm{Ass}$ be an $A_\infty$-operad in topological spaces, and let $X$ be an $A$-algebra. I see three possibilities to construct a delooping out of $X$:
Rectify $X$ by taking the ...
- 1,623
4
votes
1
answer
208
views
What should be required from a model category so that the category of algebraic objects in it has the natural model structure?
I have two reference questions
What should be required of a category with finite products so that a (multi-sorted, finitary) Lawvere theory induces a monadic adjunction on it? This should be ...
- 6,962
1
vote
1
answer
288
views
comparison between two monadic definitions for an operad
According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$
According to Leinster, an operad is ...
- 1,446