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# Questions tagged [infinity-categories]

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### Proving a Kan-like condition for functors to model categories?

I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...
149 views

### Existence of pointwise Kan extensions in $\infty$-categories

This answer by Emily Riehl mentions a to-be-published proof of the fact that $\infty$-categorical (pointwise) Kan extensions exist when the target category admits (co)limits indexed by the relevant ...
279 views

### Is there such a field as applied $\infty$-category theory?

It seems that applied category theory has exploded in popularity in recent years. My question is simple: had there been any work using $\infty$-category theory in applications? Edit: By ...
79 views

112 views

### Inner fibrations are Kan fibrations on Map sets

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

### Straightening of a marked simplicial set

Let $X$ be in $Set_{\Delta}$, let $\phi=id:\mathbb{C}X\rightarrow \mathbb{C}X$ be the map of simplicial categories over which we want to straighten. Assuming that $X$ is a quasi-category, how can one ...
42 views

### Some properness condition in simplicial sets

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 ...
540 views

### What does the homotopy coherent nerve do to spaces of enriched functors?

Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a ...

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