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[Edit: Corrected some false claims and modified questions accordingly.] Let $\mathcal{S}$ be the cocomplete $(\infty, 1)$-category generated by a point. This is conventionally known as the $(\infty, 1 …

The standard formulation of the univalence axiom for a universe type $U$ is that, for all $X : U$ and $Y : U$, the canonical map $(X =_U Y) \to (X \simeq Y)$ is an equivalence. As we (usually) cannot …

Let $\mathcal{A}$ be a category. There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf of sets" (i.e. …

I assume $\varinjlim_{j : \mathcal{J}} \mathcal{I}_j = \mathcal{I}$ is meant in the strict sense of 1-categories. Since $\textbf{Cat}$ is cartesian closed, $$\textstyle [\mathcal{I}, \mathcal{C}] \con …

The category of descent data is indeed the homotopy limit of your cosimplicial diagram. In the case where $\mathcal{F}$ actually is fibred in categories (and not higher categories), then you can trunc …

Let $\mathcal{M}$ be the following category: The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels. The morphisms are functors that preserve the struct …

We can detect the difference between the two constructions using the homotopy category. Given any simplicially enriched category $\mathcal{C}$, we can construct an ordinary category $\pi_0 [\mathcal{C …

Yes, one can use functorial factorisation (of weak equivalences with a fixed codomain) to show that two categories of spans are homotopy equivalent is correct. Actually, the only subtle point I am a …

There are several possible definitions of initial object in a 2-category $\mathfrak{K}$; which one is appropriate depends on your applications. A 2-category has an underlying ordinary category, so we …

I will answer the title question about computing homotopy limits. Let $A : \mathcal{C} \to \mathbf{Cat}$ be a small (strict) diagram. By expanding the definitions of the cobar construction, one eventu …

There is only a set of isomorphism classes of étale geometric morphisms between any two Grothendieck toposes. In fact, the same is true for essential geometric morphisms. Recall that Grothendieck top …

My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of: Cordier and Porter proved a …

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent …

There is a Grothendieck topos $\textbf{Set}[\mathbb{O}]$ with the following universal property: for all Grothendieck toposes $\mathcal{E}$, the category $\textbf{Geom}(\mathcal{E}, \textbf{Set}[\mathb …