It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to construct limits and co-limits. I'm especially interested in co-equalizers.
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1$\begingroup$ That's extremely complicated. There's no explicit way of doing it. The existence of such colimits is via general categorical results which, in my experience, are impossible to trace back in order to recover a sensible construction. $\endgroup$– Fernando MuroCommented Apr 23, 2013 at 20:26
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$\begingroup$ @Fernando is it also complicated to construct equalizers when the commutative ring is a field?. $\endgroup$– Victor TCCommented May 6, 2019 at 8:19
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$\begingroup$ @VictorTC equalizers (and limits in general) are easy. The equalizer of two maps is just the kernel of the difference. $\endgroup$– Fernando MuroCommented May 6, 2019 at 9:35
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$\begingroup$ @Fernando I apologize, I meant the coequalizer when the commutative ring is a field. $\endgroup$– Victor TCCommented May 6, 2019 at 9:54
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1$\begingroup$ @VictorTC coequalizers are very complicated in general. I don't think that any property of the ground ring could simplify things. $\endgroup$– Fernando MuroCommented May 6, 2019 at 11:16
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There is a general bit of category theory that was applied to ring spectra in EKMM ([83] on my website) and I'll refer to that for details. Unless I am missing something, the discussion surely specializes just as well to dg algebras over a commutative ring $R$. I'll outline the recipe it gives for constructing all colimits of dg $R$-algebras as reflexive coequalizers in the category $Ch_R$ of chain complexes over $R$.
The free graded $R$-algebra functor induces a free dg $R$-algebra functor on $Ch_R$. That gives a monad $T$ on $Ch_R$ whose algebras are the dg $R$-algebras. This monad preserves reflexive coequalizers by Prop. 7.2, p. 47. Therefore, by Lemma 6.6, p. 46, if $g\colon B\to C$ is a reflexive coequalizer of maps $e,f\colon A\to B$ in $Ch_R$ such that $A$ and $B$ are $T$-algebras and $e$ and $f$ are maps of $T$ algebras, then $C$ has a unique structure of $T$-algebra such that $g$ is a map of $T$-algebras, and $g$ is the coequalizer of $e$ and $f$ in the category of $T$-algebras. Now all colimits in the category of $T$-algebras are constructed from just such reflexive coequalizer diagrams in $Ch_R$, as shown in the proof that the category of $T$-algebras is cocomplete given in Prop. 7.4, p. 49.
answered Apr 24, 2013 at 3:17