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For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

26 votes
Accepted

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

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 …
Jacob Lurie's user avatar
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11 votes
Accepted

Example of a (presentable $k$-linear $\infty$-)category which is dualizable but not compactl...

If $X$ is a locally compact topological space, then $\mathrm{Shv}(X, \mathrm{Mod}_{k} )$ is a presentable $k$-linear stable $\infty$-category which is dualizable (in fact, self-dual), but is rarely co …
Jacob Lurie's user avatar
  • 17.8k
13 votes
Accepted

The universal property of the unseparated derived category

Yes, both of these statements are true (I thought they were in the book, but I can't seem to find them now). Here is a proof sketch. Let's start with the case described in 2). Let $\mathcal{C}$ be an …
Jacob Lurie's user avatar
  • 17.8k
64 votes
Accepted

What is homology anyway?

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 …
Jacob Lurie's user avatar
  • 17.8k
20 votes
Accepted

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

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 …
Jacob Lurie's user avatar
  • 17.8k
11 votes
Accepted

$(\infty, 1)$-Yoneda embedding via the Grothendieck construction

Let $\mathcal{M}$ be the simplicial set defined by the formula $Hom( \Delta^{J}, \mathcal{M} ) =Hom( \Delta^{J^{op} } \star \Delta^{J}, \mathcal{C} )$, so that an $n$-simplex of $\mathcal{M}$ is a $(2 …
Jacob Lurie's user avatar
  • 17.8k
19 votes
Accepted

Lower Algebra: Modules over the monoidal category of abelian groups

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 …
Jacob Lurie's user avatar
  • 17.8k
11 votes
Accepted

Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the...

If $\mathcal{C}$ is an $\infty$-topos, the $\infty$-category of groupoid objects of $\mathcal{C}$ is equivalent to the full subcategory of $Fun( \Delta^1, \mathcal{C})$ spanned by the effective epimor …
Jacob Lurie's user avatar
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42 votes
Accepted

Is Lemma A.1.5.7 in Higher Topos Theory correct?

Looks like a typo. Condition $(4)$ should say that $B$ is downward closed under $\leq$, not under $\preceq$ (otherwise, $Y_B$ is not defined).
Jacob Lurie's user avatar
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13 votes
Accepted

Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces?

If you don't group complete, then free $E_{\infty}$-spaces are $1$-truncated. Consequently, for $k > 0$, the answer is "all $E_{\infty}$-spaces". When $k=0$, you'll get those which are homotopy equiva …
Jacob Lurie's user avatar
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26 votes
Accepted

How much do universes matter in topos theory?

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 …
Jacob Lurie's user avatar
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20 votes
Accepted

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

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 …
Jacob Lurie's user avatar
  • 17.8k
33 votes

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

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 bicategor …
Jacob Lurie's user avatar
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20 votes
Accepted

Semi-simplicial versus simplicial sets (and simplicial categories)

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 …
Jacob Lurie's user avatar
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20 votes
Accepted

The weak equivalences in the covariant model structure

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 set …
Jacob Lurie's user avatar
  • 17.8k

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