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5 votes

Is the Grothendieck construction a homotopy pullback?

The analogy is of course a correct/useful analogy, but I think any model structure for which the literal statement is correct must have $Set_* \simeq *$, so it would be a bit too coarse to do anything ...
Maxime Ramzi's user avatar
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21 votes

Useful ideas in category theory which violate the principle of equivalence

I would think about this question in this way: If you have a construction that violates the equivalence principle then either (A) it is a strictified or simplified version of something that is ...
3 votes

Does forgetting colimits preserve colimits?

Here is an example showing that $U_{\tau, \kappa}$ does not preserve pushouts for any pair of regular cardinals $\tau > \kappa$. Let $C = \tau$, where here $\tau$ is thought of as an ordinal (i.e. ...
G. Stefanich's user avatar
4 votes

Are $\infty$-categories functorially colimits of their simplices?

This is just an expended version of the comment. The answer to the question as asked is no. The problem is that for any ($\infty$-)category $J$ the category $D_J$ of functors $J \to Cat_\infty$ that ...
Simon Henry's user avatar
  • 38.7k
1 vote

Does forgetting colimits preserve colimits?

Here's an example to play around with, with $\kappa = \omega$ and $\tau = \omega_1$ and $Vect = Vect_k$ for $k$ a finite field, look at the pushout of $Vect \leftarrow Set \to Vect$, where both ...
Tim Campion's user avatar
  • 56.7k
2 votes

resolution property and perfect stacks

A quasi compact and quasi-separated scheme has its derived category of sheaves of modules with quasi-coherent cohomology generated by perfect complexes. This is actually a theorem of Bondal and Van de ...
Leo Alonso's user avatar
  • 8,694
5 votes

From the *usual* nerve of topological categories to $\infty$-categories

The answer to Question (ii) is positive. That is to say, there is a weak equivalences between the following functors from Segal topological categories to quasicategories: the composition of the ...
Dmitri Pavlov's user avatar

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