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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 …

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 …

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 …

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 …

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 …

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 …

It is possible to prove the contractibility directly (and thereby bypass the obstruction theory arguments sketched at the end of section 3 of my paper). The statement itself is an example of an h-prin …

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 …

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 …

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 …

It seems that the first question only makes sense for marked simplicial sets $X$ over $S$ where every edge of $X$ is marked (otherwise, the slice category is not equivalent to marked simplicial sets o …

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 …

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

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 …

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 …