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 ...

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 $x^{... View answer Accepted answer 96 votes I don't think I have a compelling answer to this question, but maybe some bits and pieces that will be helpful. One point is that all of the examples that you bring up are related to the first: ... View answer Accepted answer 59 votes Let's take coefficients in a field$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$is a space for ... View answer 45 votes Though it is not true in general, it is true whenever$Y$is separated. The map$f$from$X$to$Y$factors as a composition $$X \stackrel{f'}{\rightarrow} X \times Y \stackrel{f''}{\rightarrow} Y$$ ... View answer Accepted answer 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). View answer 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)-... View answer Accepted answer 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 ... View answer Accepted answer 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 ... View answer 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 ... View answer 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 ... View answer Accepted answer 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 ... View answer Accepted answer 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 ... View answer 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 ... View answer 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 ... View answer Accepted answer 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 ... View answer Accepted answer 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 ... View answer 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 ... View answer Accepted answer 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$...

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 $\mathcal{C}$ ...

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 ...

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 ...

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 ...

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 ...

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 ...

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.)

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 $1$-...

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) ...

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 $\mathcal{X}$ ...
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 ...