5
$\begingroup$

Marc's answer to my previous question gives a way to compute colimits in the category of presentable $\infty$-categories and continuous functors, using the (discontinuous) right adjoints to those functors. But in particular it is not true that the forgetful functor from presentable $\infty$-categories and continuous functors, to all $\infty$-categories and functors, preserves all colimits.

Does it preserve all filtered colimits? I am sorry if the answer is easy to find in Lurie's textbook. I wasn't able to do so right away.

$\endgroup$
14
$\begingroup$

It doesn't. For instance, the $\infty$-category of spectra is the colimit of the tower

$$ \mathcal{S}_* \stackrel\Sigma\to \mathcal{S}_* \stackrel\Sigma\to ... $$

in $Pr^L$, but its colimit in $Cat_\infty$ is the Spanier-Whitehead category whose objects are formal desuspensions $\Sigma^{-n}X$ of pointed spaces. The latter category is stable and (uncountably) accessible but lacks some infinite colimits, like the sum of $\Sigma^{-n}S^0$ for $n\geq 0$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.