Jacob Lurie
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If I want to study Jacob Lurie's books "Higher Topoi Theory", "Derived AG", what prerequisites should I have?
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185 votes

To read Higher Topos Theory, you'll need familiarity with ordinary category theory and with the homotopy theory of simplicial sets (Peter May's book "Simplicial Objects in Algebraic Topology" is a ...

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Hirzebruch's motivation of the Todd class
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102 votes

Since you mention playing around with residues, I'm probably not telling you anything you don't already know. But there is a systematic way to extract the power series $f$ from the coefficients of $x^{...

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Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
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96 votes

I don't think I have a compelling answer to this question, but maybe some bits and pieces that will be helpful. One point is that all of the examples that you bring up are related to the first: ...

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What is homology anyway?
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59 votes

Let's take coefficients in a field $k$, for simplicity. On 2): the singular cohomology of a topological space $X$ is the dual of its singular homology, almost by definition. But if $X$ is a space for ...

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Are morphisms from affine schemes to arbitrary schemes affine morphisms?
45 votes

Though it is not true in general, it is true whenever $Y$ is separated. The map $f$ from $X$ to $Y$ factors as a composition $$ X \stackrel{f'}{\rightarrow} X \times Y \stackrel{f''}{\rightarrow} Y$$ ...

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Is Lemma A.1.5.7 in Higher Topos Theory correct?
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39 votes

Looks like a typo. Condition $(4)$ should say that $B$ is downward closed under $\leq$, not under $\preceq$ (otherwise, $Y_B$ is not defined).

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What's the current state of the classification of not-fully-extended TQFTs?
39 votes

When n > 1 the paper that you cite can give you a little bit of traction: the sketch proof of the main result gives a generators-and-relations presentation of (k,k+1,...,k+n)-Bord relative to (k,k+1)-...

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What's so special about the forgetful functor from G-rep to Vect?
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35 votes

If $G$ is an affine algebraic group (for example a finite group), then the category of $k$-linear cocontinuous symmetric monoidal functors from $\mathsf{Rep}(G)$ to $\mathsf{Vect}_k$ is equivalent to ...

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What is a simplicial commutative ring from the point of view of homotopy theory?
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33 votes

I don't know a really satisfying answer to this question, but here are a few observations. 1) The $\infty$-category of simplicial commutative $k$-algebras is monadic over the $\infty$-category of ...

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Reflection principle vs universes
30 votes

I'm going to go out on a limb and suggest that the book HTT never uses anything stronger than replacement for $\Sigma_{15}$-formulas of set theory. (Here $15$ is a randomly chosen large number, and ...

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What is a symmetric monoidal $(\infty,n)$-category?
30 votes

There are many (equivalent) definitions for the notion of symmetric monoidal $(\infty,n)$-category. One approach is based on the observation that a monoidal category can be identified with a ...

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Relation between topos and $\infty$-topos
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27 votes

For any space $X$, there's an $\infty$-topos of spaces fibered over $X$. The underlying ordinary topos is the category of representations of the fundamental groupoid of $X$. So if $X$ is simply ...

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What do cohomology operations have to do with the non-existence of commutative cochains over $\mathbb{Z}$?
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27 votes

Via the Dold-Kan correspondence, the category of cosimplicial abelian groups is equivalent to the category of nonpositively graded chain complexes of abelian groups (using homological grading ...

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Reflection principle vs universes
26 votes

Reflecting on Gabe's comment on my original answer, I now think what I wrote is misleading because it conflates two separate (but related) assertions: The existence of strongly inaccessible cardinals ...

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What are simplicial topological spaces intuitively?
25 votes

If $\mathcal{A}$ is an abelian category, then the Dold-Kan correspondence supplies an equivalence between the category of simplicial objects of $\mathcal{A}$ and the category of nonnegatively graded ...

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How much do universes matter in topos theory?
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24 votes

The change-of-universe construction is faithful but not full. For example, let X be the topos of sets and let Y be the classifying topos for abelian groups. The category of geometric morphisms from X ...

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Stable homotopy category and the moduli space of formal groups
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23 votes

One useful thing to keep in mind is that the cohomological functor from the stable homotopy category to the category of quasi-coherent sheaves on the moduli stack $\mathcal{M}$ is not essentially ...

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Why study simplicial homotopy groups?
22 votes

If $X$ is a simplicial set which is not Kan, you can compute the homotopy groups of $X$ by choosing a weak homotopy equivalence $f: X \rightarrow Y$ where $Y$ is Kan and then applying the construction ...

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How do $\infty$-categories allow us to do descent on the derived level?
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21 votes

Let $X$ be a topological space covered by open sets $U$ and $V$. Let $\mathscr{F}$ and $\mathscr{G}$ be complexes of sheaves defined on $U$ and $V$, respectively. Suppose you are given an isomorphism $...

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Functorial kernel in derived category
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20 votes

Let $\mathcal{C}$ be a stable $\infty$-category. Then $\mathcal{C}$ has a homotopy category $h \mathcal{C}$, which is triangulated. The collection of morphisms $f: X \rightarrow Y$ of $\mathcal{C}$ ...

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Examples of $(\infty,1)$-topoi that are not given as sheaves on a Grothendieck topology
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20 votes

Marc's examples are good ones, but let me add two more (which are closely related to each other): 1) Let $\mathcal{C}$ be an accessible $\infty$-category which admits small filtered colimits, and let ...

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The weak equivalences in the covariant model structure
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20 votes

Maybe it would be helpful to think about the analogous situation in ordinary category theory. Suppose you are given a category $\mathcal{E}$ and a functor $F$ from $\mathcal{E}$ to the category of ...

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Lower Algebra: Modules over the monoidal category of abelian groups
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19 votes

A locally presentable category $\mathcal{C}$ has a (unique) structure of an $Ab$-module if and only if it is additive. Such a category need not be abelian. This is one reason to prefer the setting of ...

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What would be an infinity-groupoid analogue of the duality between sets and complete atomic boolean algebras?
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19 votes

Let $\mathcal{S}$ denote the $\infty$-category of spaces. For any $\infty$-topos $\mathcal{X}$, there is an essentially unique geometric morphism $\pi^{\ast}: \mathcal{S} \rightarrow \mathcal{X}$. The ...

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compact objects in model categories and $(\infty,1)$-categories
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19 votes

If $\mathcal{C}$ is a combinatorial model category, then for all sufficiently large regular cardinals $\kappa$, an object of the underlying $\infty$-category is $\kappa$-compact if and only if it can ...

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Where to find the correct result in Higher Algebra, incorrect reference
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18 votes

The correct reference is 6.1.4.14. (And the hypothesis of 6.1.6.27 should refer to countable limits and colimits, rather than finite limits and colimits.)

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Are there noncommutative extensions of $\alpha_p$ by $\mathbb{G}_m$?
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18 votes

So, bilinear maps $\alpha_p \times \alpha_p \rightarrow \mathbf{G}_m$ are classified by maps from $\alpha_p$ to itself (since $\alpha_p$ is Cartier self-dual). The collection of such maps is a $1$-...

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elliptic curves and group cohomology
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17 votes

As Charles indicates, "the moduli stack of $G$-bundles on $E$" is not quite the right thing to consider, especially if you're not working over $\mathbf{C}$. This is for two (unrelated) reasons: 1) ...

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The most general context of Mather's Cube Theorems
17 votes

Let $\mathcal{X}$ be an $\infty$-category (i.e., a homotopy theory) which admits small homotopy colimits, a set of small generators, and has the property that homotopy colimits in $\mathcal{X}$ ...

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Semi-simplicial versus simplicial sets (and simplicial categories)
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17 votes

The map $j_! j^{\ast} K \rightarrow K$ is never a Joyal equivalence unless $K$ is empty. For example, if $K = \Delta^{0}$, then $j_{!} j^{\ast} K$ is the nerve of the category with one object $X$ and ...

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