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3 votes
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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
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3 votes
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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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4 votes

“Geometric” vs Homotopical completion

In the affine case, this is more-or-less proved in [Bhargav Bhatt, Completions and derived de Rham cohomology]. More precisely, Kathryn Hess' completion is more akin to Carlsson's Adams completion, ...
Z. M's user avatar
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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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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 ...
1 vote
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Simplicial enrichment on unbounded algebras over an operad

There is no obstruction. If $M$ is a simplicial monoidal model category, and $O$ is an operad in $M$, then the category of $O$-algebras is simplicially enriched, tensored, and cotensored. If it's a ...
David White's user avatar
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2 votes
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“Geometric” vs Homotopical completion

Yes, there's a way to relate the two. First, it's helpful to think of both in terms of universal properties. Since geometric completion is a fiber product, it's a pullback. Meanwhile, the homotopical ...
David White's user avatar
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1 vote

Homotopical Combinatorics

Back when this question was asked, several people suggested they were not sure what was meant by "homotopical combinatorics." Well, now there's a subfield known as homotopical combinatorics. ...
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

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

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
2 votes

Reference request for equivalences between different models of lax limits

This is a great question. Let me start with limits and discuss lax limits later. Given a $D$-shaped diagram $X$ of model categories (where $D$ is a small category), one can ask whether the two ways (...
David White's user avatar
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3 votes
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Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection

This sounds like a very exciting phase of your studies! As you point out, there's plenty of intersections between algebraic geometry and homotopy theory. One intersection is motivic homotopy theory, ...
David White's user avatar
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13 votes
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Plus construction on Simplicial Sets?

The answer is yes. This is spelled out in the book The local structure of algebraic K-theory by Bjørn Ian Dundas, Thomas G. Goodwillie and Randy McCarthy. Check out Section 1.6.1 on page 26, where ...
David White's user avatar
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3 votes
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Is the Whitehead bracket $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$ injective?

The answer is (still) no. It comes down to the assertion that the homomorphism $\pi_p(A)\otimes \pi_q(B)\to \pi_{p+q}(A\wedge B)$ is not always injective. To see this, let us rewrite the Whitehead ...
Gregory Arone's user avatar

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