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Yes, all the colimits that exist in ${\cal K}$. Indeed, ${\cal K}_{{\rm fin}}$ can be identified with the smallest full subcategory of ${\rm Fun}({\cal K},{\rm Set})^{{\rm op}}$ which contains the rep …

I'm not sure about the linear case described in Theo's answer, but in the setting of locally presentable categories there are dualizable objects which are not presheaf categories. Instead they are non …

Let us fix a universe and use the words "large" and "small" with respect to that universe. Presentable $\infty$-categories are typically large $\infty$-categories (since, as the previous answer mentio …

Here is a setting in which the question can be made more precise. Consider a continuous map $f: X \to Y$ of topological spaces which is also open. Let us denote by $O(X)$ and $O(Y)$ the posets of open …

Given two model categories $\mathcal{M},\mathcal{N}$, one does know what would have been a left Quillen functor out of what would have been the tensor product $\mathcal{M} \otimes \mathcal{N}$ into a …

There is another simple kind of peusdo-limits called inverters, in which one universally inverts a 2-morphism between two parallel 1-morphisms (such pseudo-limits can also be constructed in a simple w …

For the 1-categorical case, it seems to be indeed a question of soundness. More precisely, the condition that $P_{\cal I}({\cal C}) = P^{{\cal K}}({\cal C})$ is equivalent to the condition that every …

If $\mathcal{C}$ is a small category with finite limits then geometric morphisms from ${\rm Set}$ to the presheaf topos ${\rm PSh}(\mathcal{C})$ are in bijection with left exact functors $\mathcal{C} …

I think there is a typo in Lemma 5.4.5.11: $K$ is supposed to be $\tau$-small and not $\kappa$-small. Note that if $\tau < \kappa$ and $K$ is $\kappa$-small but not $\tau$-small then the statement of …

The question that you ask might be better phrased intrinsically without referring to sites. Fix an $0 \leq n \leq \infty$ and let $\mathbf{X}$ be an $n$-topos. Recall that an object $X \in \mathbf{X}$ …

You can argue as follows. Suppose that $g: D \to D'$ is a retract of $f: C \to C'$ (in the category of $S$-enriched categories) via maps $D \stackrel{i}{\to} C \stackrel{r}{\to} D$ and $D' \stackrel{i …

It might be worth to first consider the particular case of the symmetric monoidal category $({\rm Set},\times)$ of sets and Cartesian products. Let us mildly extend the setting to include possibly mul …

Here is also a simple example where the restriction of $F$ to the heart is faithful but $F$ itself is not. Let $Vec$ be the abelian category of complex vector spaces and let $Rep(\mathbb{Z})$ be the a …

Specifically for the case of quasi-categories (or any other model for $\infty$-categories) the following observation can be useful: suppose that $f: {\cal C} \to {\cal D}$ is a map of quasi-categories …

Fiber products do commute with sequential colimits of closed embedding in CGWH, but one must remember that neither limits nor colimits in CGWH are the same as the corresponding limits and colimit in t …