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8
votes
0answers
235 views

From the perspective of bordism categories, where does the ring structure on Thom spectra come from?

To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...
3
votes
0answers
147 views

matrix ring spectra

I am trying to understand matrix ring spectra. Apparently, I have two different definitions of those and I did not manage to show that they are equivalent - maybe they even are not in the general ...
5
votes
1answer
201 views

endomorphisms of modules over symmetric ring spectra

I have a probably very basic question about modules over symmetric ring spectra: Let $R$ be a commutative symmetric ring spectrum and let $M$ and $N$ be module spectra over $R$. Moreover, let ...
12
votes
2answers
368 views

Truncations of E_infinity algebras

In section 4.1 of Lurie's DAG VIII, he implies the existence of an $E_\infty$-ring spectrum $A$ such that the coconnective truncation $\tau_{\leq 0} (A)$ does not admit the structure of an ...
1
vote
0answers
171 views

Units of a ring spectrum

Is there a good notion of the spectrum of units $R^\ast$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$? A standard definition (see section 1.2 in http://arxiv.org/abs/0810.4535) seems ...
20
votes
1answer
967 views

Why not a Roadmap for Homotopy Theory and Spectra?

MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up ...
4
votes
1answer
504 views

Can motivic E_∞-ring spectra be strictified to commutative motivic symmetric ring spectra?

Theorem 4.4.4.7 in Lurie's Higher Algebra (or Theorem 4.3.22 in DAG III) states (roughly speaking) that under certain conditions the ∞-category of commutative ∞-monoids in a given symmetric monoidal ...
4
votes
1answer
464 views

Topological Hochschild cohomology?

Let $A$ be a $E_\infty$-ring spectrum. By EKMM, it may be treated as a commutative algebra in the appropriate category. In particular, one may define topological Hochschild homology as ...
2
votes
1answer
139 views

Does Wolbert's derived equivalence between $E_*^R$-local $R$-modules and $R_E$-modules come from a Quillen equivalence?

Let $R$ be a ring spectrum (in the world of EKMM $S$-modules) and let $E$ be a smashing $R$-module. Denote by $R_E$ the $E_*$-localization of $R$. By a theorem of Wolbert (Theorem 2 in Classifying ...
5
votes
1answer
111 views

Is this a description of the $\aleph_1$-localizing subcategory generated by a compact generator?

This should be obvious but I'm not seeing it: The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\_)$ preserves ...
7
votes
2answers
308 views

Computing a cobordism group of manifolds endowed with a real vector bundle with constraints on the Stiefel-Whitney classes

I am interested in computing the cobordism group of oriented manifolds $M$ of dimension 7 endowed with real vector bundles $N$ of rank 5 with the following conditions on the Siefel-Whitney classes: $ ...
1
vote
0answers
90 views

Relating the cardinalities of rings and their spectra

Does there exist an uncountable (possibly commutative, unital) ring with a countably infinite spectrum? More generally, given two cardinalities $\kappa$ and $\lambda$, does there exist a ring $R$ ...
7
votes
1answer
575 views

Are Thom spectra MU, MSO and K-theory spectra KU, KO modules over some truncations of the sphere spectrum?

The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum. In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO). On the other hand, MU and ...
11
votes
1answer
295 views

On the stable splitting of loops on a suspension

Let $X$ be a connected, based CW complex. Then the James splitting of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra $$ \Sigma^{\infty} \Omega\Sigma X_+ \quad \simeq \quad ...
3
votes
1answer
220 views

If a t-truncation of the unit object in a stable homotopy category is a ring object up to homotopy, can it be lifted to a ring spectrum? What about the Postnikov t-truncations of the sphere spectrum?

Let $S$ be the unit object in a monoidal stable homotopy category $SH$ (we demand that the multiplication $S\times S\to S$ is commutative and associative on the level of spectra, and not just up to ...
6
votes
1answer
237 views

State of knowledge on the Commutative W-spaces which appear in “Model Categories of Diagram Spectra”

This is a follow-up question to another question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and prove monoidal Quillen ...
2
votes
2answers
360 views

Orientation of complex bordism spectrum

I have the following question: If $E$ is a ring spectrum, then a complex orientation of $E$ is an element of $E^2(\mathbb{C}P^{\infty})$ that is mapped to $1$ in $E^2(\mathbb{C}P^{1})$. I have read ...