In F. Waldhausen's paper "Algebraic K-theory of generalized free products, Part I", page 142,line 19, there is a term "order two sequence". Can anyone explain its meaning to me? According to the context, it should be a generalization of short exact sequence. Thanks!
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2$\begingroup$ Dear @Kun Wang: I think it would be very helpful if you were to copy the relevant text from Waldhausen's article into the question. All other things being equal, a question which is clear and self-contained is always easier to answer. $\endgroup$– Ricardo AndradeCommented Dec 2, 2013 at 0:46
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1$\begingroup$ Thanks! Here is a link to the article:pub.uni-bielefeld.de/luur/… $\endgroup$– Kun WangCommented Dec 2, 2013 at 0:50
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An "order two sequence" is a sequence of morphisms such that the composition of any two consecutive morphisms is the zero morphism. At the point you are asking about in Waldhausen's paper, the condition is then $\kappa \circ \iota=0$ (which always holds by the definition of completed splitting diagram).
References:
[Barry Mitchell, Theory of Categories, Section 15]
[Marco Grandis, On the categorical foundation of homological and homotopical algebra, Section 1.2]
answered Dec 8, 2013 at 21:36