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Let $D : I \to C$ be a diagram in a category $C$ (i.e. $D$ is a functor and $I$ is small), $x \in C$ and $x \to D$ a cone in $C$. Is there a notion of a split cone? If $x \to D$ is a split cone, then $x \to D$ should be an absolute limit cone, i.e. for every functor $F : C \to C'$ the cone $F(x) \to F \circ D$ is a limit cone in $C'$ (but not vice versa). The idea behind this notion should be that only some equations have to be checked, instead of checking a universal property for all test objects in $C$.

This should generalize the notion of a split fork, which is the case $I = \bullet ~ {\longrightarrow\atop\longrightarrow}\bullet$. I am also interested in special cases of $I$. For example, when $I$ is finite and discrete, we have - at least for $\mathsf{Ab}$-enriched categories - the notion of a biproduct: A diagram $\{P \xrightarrow{p_n} A_n\}_{n \in I}$ is a biproduct if there are morphisms $i_n : A_n \to P$ satisfying $p_n i_m = \delta_{n,m} \mathrm{id}$ and $\sum_n i_n p_n = \mathrm{id}$.

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    Huh, apparently I was mistaken. See the fourth bullet point [here](http://ncatlab.org/nlab/show/absolute+colimit#particular_absolute_colimits_15).2012-12-21

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