In remark 7.1.0.4. of Lurie's Higher Algebra, the sequence $(E_n^{\otimes})_{0\leq n\leq\infty}$ of $\infty$-operads induces forgetful functors for the sequence of categories $(\operatorname{Alg}^{(n)}(\mathrm{Sp}))_{0\leq n\leq\infty}$ of $E_n$-rings. Do these forgetful functors preserve limits and colimits?
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4$\begingroup$ Forgetful functors preserve small limits and sifted colimits. Not all small colimits, e.g. they do not preserve finite coproducts. $\endgroup$– Z. MCommented Feb 7 at 12:53
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$\begingroup$ Thank you very much for your answer. Could you give a reference for this result? $\endgroup$– MisoCommented Feb 7 at 14:34
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1 Answer
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Limit-preservation is the content of section 3.2.2 of Higher Algebra (see particularly Corollary 3.2.2.5). Preservation of sifted colimits is in section 3.2.3 (see particularly Corollary 3.2.3.2).
As Z. M points out, coproducts are not preserved. This is parallel with ordinary algebra: the coproduct of $\Bbb Z[x]$ and $\Bbb Z[y]$ in the category of commutative rings is $\Bbb Z[x,y]$, whereas the coproduct in the category of associative rings is the noncommutative polynomial algebra $\Bbb Z\langle x,y\rangle$.
answered Feb 7 at 14:50
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1$\begingroup$ Thank you for your detailed answer with examples! I will try to understand these results with your hints. $\endgroup$– MisoCommented Feb 7 at 15:11