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I have a question related to the discussion (Coequalizer in category of dg-algebras). How do you prove that the category of (small) dg-categories and the category of (small) A_{\infty} categories are complete and cocomplete? I am particulary interested in understanding how to compute limits and colimits of A_{\infty} categories and see if it is possible to have an explicit description of such.

Thank You.

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    $\begingroup$ More or less the same. They are categories of algebras over an accessible monad in a locally presentable category. On objects, (co)limits are computed as in the category of sets. On morphisms, limits and filtered colimits are computed as in the category fo chain complexes. Other colimits, such as push-outs and coequalizers, are very complicated in general. $\endgroup$
    – Fernando Muro
    Commented Jan 8, 2014 at 15:18
  • $\begingroup$ Thanks, but something is not clear to me. According to (mathoverflow.net/questions/127028/…) the category of small A_{\infty} categories does not have all (small) limits. This seems to happen already at the level of A_{\infty} algebras, for instance see Lefevre-Hasegawa, where a "model structure without limits" is defined. Also, the inclusion of dg-categories in A—{infty} categories is a right adjoint (the enveloping dg-category is its left adjoint). So colimits of dg-categories differ in the two settings. $\endgroup$
    – Giovanni Faonte
    Commented Jan 10, 2014 at 15:54
  • $\begingroup$ Giovanni, I converted your answer to a comment (where it belongs), but had to edit it to make it fit. Generally we try to use answer boxes only for actual answers to questions, not follow-up comments/questions. $\endgroup$
    – Todd Trimble
    Commented Jan 10, 2014 at 16:03
  • $\begingroup$ Oh sorry, I am somewhat new to this website. $\endgroup$
    – Giovanni Faonte
    Commented Jan 10, 2014 at 18:02
  • $\begingroup$ Giovanni, what morphisms are you thinking of? $\endgroup$
    – Fernando Muro
    Commented Jan 10, 2014 at 22:12

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