I am wondering about the following two related questions and don't know if they have already clear answers or not.

1) Suppose that we already know the functor $F \colon \mathcal{C} \to \mathcal{D}$ preserves limits or colimits. Moreover we know that it sends crossed modules to crossed modules. Thus can we say that, this functor also preserves limits or colimits of crossed modules as well?

2) Suppose that we have adjoint functors $F \colon \mathcal{C} \to \mathcal{D}$ and $G \colon \mathcal{D} \to \mathcal{C}$ which preserve crossed module structures. Being adjoint functors implies preserving limits or colimits depending right/left issue, right. But the question is, can we also say that these two functors preserve limits or colimits again of crossed modules? Or can we extend it simply to adjunction of crossed module functors?

PS: Categories are arbitrary algebraic categories where crossed module notion already exists.

Thanks in advance,

  • $\begingroup$ Is $\mathcal{C}$ a specific kind of ctegory? Otherwise, could you please explain what is a cross module in an abstract category? $\endgroup$ – Uri Bader Apr 22 '16 at 11:47
  • $\begingroup$ I edited the question with the point of your comment, thanks! $\endgroup$ – Kadir Emir Apr 22 '16 at 11:53

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