# cokernel for $L_\infty$-algebra morphisms

As I have asked a wrong question previously, I edited a bit.

It is obvious that the cokernel construction does not work well for the category of Lie algebras. The cokernel exists only for a normal inclusion $\mathfrak{h}\hookrightarrow \mathfrak{g}$.

It is said that 2-categories(=bicategories) have a weak notion of limits/colimits. In some cases, even if a category $C$ does not have limits, but there may exist a way to view/extend, what ever, $C$ as a 2-category in which the limits work well.

Therefore, I am wondering

(1) Whether there is such a 2-categories for Lie algebras in which 2-categorical cokernels work well. I know there is a notion of Lie 2-algebras(and $L_\infty$-algebras), but I do not know whether the cokernel of a Lie algebra morphism, other than normal inclusions, viewed as a Lie 2-algebra morphisms exists.

(2) Whether there is a universal way to fix a cokernel(or other limits/colimits) ill-behaved category, obtaining a category(maybe 2-category)?

Thanks to commenters, I realized that the naive statement on Lie algebras is simply wrong. My original attempt is to consider the cokernel for $L_\infty$-algebra morphisms. My second question still make sense.

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Can you say what it would mean for cokernels to "work well"? The category of Lie algebras has all colimits, so there is a sense in which cokernels "work". – Tom Leinster Dec 2 '11 at 17:36
@Tom Leinster Good to know that, could you tell me the cokernel of a general Lie algebra morphism $f:\mathfrak{h}\to \mathfrak{g}$? Is it $\mathfrak{g}$ quotient by the ideal spanned by $\im f$? – Ma Ming Dec 3 '11 at 1:12
Yes, the cokernel of a morphism is the quotient by the ideal generated by the image. – Fernando Muro Dec 3 '11 at 1:34
2-categories are not the solution, I believe, but model categories (which are sometimes thought of as $\infty$-categories). Lie algebras do not form a model category,, but differential graded Lie algebras do, and they contain classical Lie algebras. Moreover, they are equivalent to $L_\infty$-algebras. In model categories, cokernes are replaced by mapping cones, which are 'homotopy cokernels'. – Fernando Muro Dec 4 '11 at 11:21
@Fernando Muro Yes, I am seeking the homotopy cokernel/mapping cone of L infinity algebras. – Ma Ming Dec 12 '11 at 17:05