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While studying a completely unrelated problem, I have proved something on the following line: given a diagram of simplicial sets $X: C \to \textrm{sSet}$, some deformations of the geometric realizatio …

In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant …

Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ satisfies som …

Let $F: C \to D$ be an infinity functor. Is it true that the homotopy fiber at $y$ can be described as $C \times_D D^{\simeq}_{/y}$? If not, is there a simple formula resembling this one? Beside the i …

Simple case. Take $X_{\bullet}$ a cosimplicial space. Is it true that the chain complex of its homotopy totalization is quasi-isomorphic to the homotopy totalization of its chain complex? Because of t …

Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\inft …

It is an elementary fact that a map $f:X \to Y$ between spaces induces a chain complex morphism $f_* : C_*(X) \to C_*(Y) $. This allows one to transfer 1-category theoretic arguments from spaces to as …

I have been a bit sloppy in the title, but let me be specific. I stepped again into the subtle difference between homotopy limit and limit in the homotopy category, in the following version. Suppose y …

In remark 1.2.6.2 (HTT), Lurie states that Another possible approach to the problem of homotopy coherence is to restrict our attention to simplicial (or topological) categories C in which every homot …

Let $tr_2: \mathrm{sSet} \to \mathrm{sSet}_{\le 2} $ be the 2-truncation functor. Let $C$ be a 2-truncated simplicial set such that every horn $tr_2( \Lambda^2_1) \to C$ extends to $tr_2(\Delta_2) \ …

I am exploring a bit the world of groupoids. What I have in mind is that infinity groupoids correspond to spaces. So my first question is the following: Consider the model category $\infty-Grpd$ of …

I am trying to reduce the computation of weak operadic colimits to colimits. Let me introduct some notation. Let $q:C^{\otimes} \to N(Fin_*)$ be a symmetric monoidal category. Let $p: K \to C^{\otime …

Firstly, a bit of notation. Let $C$ be a simplicial set. We define, for $x,y \in C$ vertices in $C$ $$Map(x,y) = \{x\}\times_{\Delta^{\{0\}}}Map_{sSet}(\Delta^1,C) \times_{\Delta^{\{1\}}} \{y\} $$ t …

Suppose that $C \to D$ is a trivial Kan fibration, and $D' \to D$ is an equivalence of simplicial sets. Let $C'$ be their pullback. Is it true that $C' \to D'$ is an equivalence? Recall that a trivia …

I am studying homotopical cosheaves, and I came up with the following "conjecture". We can see an "additive" precosheaf in chain complexes (such that corestrictions do not commute on the nose) as a …