# Cocompleteness of the category of small $A_\infty$ categories

To follow up on my previous question, is the category of small $A_\infty$ categories even cocomplete? Looking for reference.

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The 1-category of small $A_\infty$ categories is certainly cocomplete, as it is the category of representations of a Lawvere algebraic theory and hence presentable. But this is, of course, not the category that you care about. Presumably, you care instead about some version of an $(\infty,2)$-category, since $A_\infty$ categories are some version of $(\infty,1)$-categories. –  Theo Johnson-Freyd Apr 20 '13 at 5:43
I actually do want the plain 1-category. Could you please give a reference? Sounds very interesting. –  yasha Apr 20 '13 at 15:00