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Higher category theory is, roughly speaking, where category theory meets homotopy coherent mathematics. It is hence relevant to those problems in which categorical structures and homotopy coherent phe …

1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety i …

I believe that this is a particular case of Lurie's "operadic left Kan extension". We may identify a monoidal $\infty$-category $\mathcal{C}$ with a coCartesian fibrations of $\infty$-operads $\mathca …

Let ${\cal C}$ be the category of finite dimensional ${\bf Z}/2$-vector spaces equipped with a ${\bf Z}/2$ action, let ${\cal C'}$ be the category of finite dimensional ${\bf Z}/2$-vector spaces and l …

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 …

$\Delta_+$ is the monoidal category generated from the associative operad, considered as a non-symmetric operad. Similarly, $(\Delta_+)_{{\rm sym}}$ is the symmetric monoidal category generated from t …

One way to construct the duality functor ${\cal C} \to {\cal C^{\rm op}}$ is through the notion of a pairing of $\infty$-categories (see HA, Definition 5.2.1.5). In particular, in this case we're talk …

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 $X \in \mathbf{X}$ …

The (epi,mono) factorization system in Sets is part of a model structure on Sets whose weak equivalences are the epis, fibrations are monos and cofibrations are everything. This is a model for the hom …

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} …

The paper "A distinguishing example in k-spaces" by John Isbell constructs an example of a locally compact space $X$ which is not compact-Hausdorffly generated.

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 …

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 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 …

Concerning the first question: the simplicial localization functor $L^H$ induces an equivalence from the relative category of small relative categories to the relative category of small simplicial cat …