10 votes

Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?

In the ten years since this question was asked, there has been a lot of progress in algebraic $K$-theory. For example, Achim Krause, Ben Antieau, and Thomas Nikolaus came up with an algorithm to ...
David White's user avatar
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8 votes

Homotopy theory and algebraic topology last 10 years. Is it a dying field?

No, it's not dying at all. If anything, now is the best time to do homotopy theory. Thanks to the recent work of Lurie and others, homotopy theory is easier than ever to get into (advances have ...
6 votes

Homotopy theory and algebraic topology last 10 years. Is it a dying field?

If the criterion is “results using algebraic topology which shocked the mathematical community in the last 10 years”, then how about Abouzaid and Blumberg’s proof of the Arnol’d conjecture using ...
6 votes
Accepted

Identifying two definitions of orientation on a vector space

Here's a direct way to relate the two: One more structure is additivity of orientations. For $V, W$ of dimensions $n,m$, we have a canonical pairing $$ \Lambda^n V \otimes \Lambda^m W \cong \Lambda^{n+...
Achim Krause's user avatar
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6 votes

Bar construction in commutative algebras is calculated by pushout

A way to see this which doesn't dive into the specifics of the simplicial diagram "$C\otimes D^{\otimes n}\otimes E$" is to apply 3.2.4.7 to the symmetric monoidal $\infty$-category $Mod_D(\...
Maxime Ramzi's user avatar
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6 votes

Bar construction in commutative algebras is calculated by pushout

Welcome to MathOverflow! First, let me point out that what you're asking is already true at the 1-categorical level. The pushout in the category of commutative rings is computed by the tensor product. ...
David White's user avatar
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5 votes
Accepted

$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$

If $A$ is a braided ∞-group, the delooping $\def\B{{\sf B}}\B A$ is an ∞-group. Consider the ∞-category of spaces equipped with an action of the ∞-group $\B A$. Since $\B Ω G≃G$, this ∞-category is ...
Dmitri Pavlov's user avatar
4 votes
Accepted

Extending curves on a surface to a basis for its first homology satisfying intersection criteria

If I understand your question correctly, what you’re looking for is Lemma A.3 in my paper here.
Andy Putman's user avatar
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3 votes
Accepted

Bar construction in commutative algebras is calculated by pushout

This is just to be explicit about the role of the bar construction in David's answer. If $I$ is the category $a \leftarrow b \rightarrow c$ parametrizing pushout diagrams, then there is a functor $f: ...
Tyler Lawson's user avatar
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3 votes
Accepted

What is the group completion of finite sets with respect to cartesian product?

As already addressed in the comments: Group completing the groupoid of finite pointed sets under the smash product gives a contractible space. The groupoid of finite sets under the cartesian product ...
Tyler Lawson's user avatar
  • 51.1k
3 votes

Does coproduct preserve cohomology in differential graded algebra category

The coproduct in the category of non-unital dg algebras is maybe easier to think about. Indeed note that the two relations in Jardine's note only have to do with the units in the two algebras. The non-...
Dan Petersen's user avatar
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2 votes

Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces

I certainly respect going back to primary sources. But, in this case, it's helpful to remember that a LOT has been written about bisimplicial sets since Quillen's 1973 paper. For example, a reference ...
David White's user avatar
  • 29.4k
2 votes

Is cohomology with local coefficients a representable functor?

Floris van Doorn's Ph.D. thesis On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory gives a very nice account of parametrized cohomology (which generalizes cohomology with ...
ಠ_ಠ's user avatar
  • 5,933
2 votes

Vector bundles over a homotopy-equivalent fibration

As indicated in the comments, this question ended up being accidentally rather trivial. Specifically, the following three facts are fairly well-known and rather easy to establish: For homotopic ...
Bence Racskó's user avatar
2 votes
Accepted

(Derived category of) sheaves over an infinite union

An easier argument would be that $R^q\pi_\ast \mathbf Q$ vanishes for $q\notin \{0,3\}$ and has rank $1$ for $q \in \{0,3\}$. Indeed, just check this on stalks. Then the cone of $\mathbf Q\to R\pi_\...
Dan Petersen's user avatar
  • 39.2k
1 vote

Comparing Kummer maps to étale homotopy at finite level

There's something crucial you're missing about the bottom-right arrow. This sends a map from $X$ to $\mathbb G_m$ to the pullback along this map. But the pullback of what? It must be the pullback of ...
Will Sawin's user avatar
  • 135k

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