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In A Course in Homological Algebra, Hilton and Stammbach.

Can anyone please explain me this notation $\{\varphi, \psi\}$ in the theorem:

Proposition 9.3: Given

$$A\xrightarrow{\{\varphi, \psi\}} B\oplus C\xrightarrow{ \left<{\gamma, \delta}\right>} D$$ we have $\left<{\gamma, \delta}\right>\{\varphi, \psi\}=\gamma\varphi+\delta\psi$.

Similarly in the Proposition 9.1.

I Don't find the notation in the book.

Thanks you all.

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  • $\begingroup$ I think it's simply the map $a\mapsto (\varphi(a), \psi(a))$ $\endgroup$ – Quang Hoang Sep 20 '15 at 7:55
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The map $\{ \varphi, \psi \}$ should be defined similarly to $\langle \gamma, \delta \rangle$ which is defined in the proof of Proposition 9.1. Let $i_1 \colon B \to B \oplus C$ and $i_2 \colon C \to B \oplus C$ be the natural injections. For $\varphi \colon A \to B$ and $\psi \colon A \to C$ the map $\{ \varphi, \psi \} \colon A \to B \oplus C$ is defined by $$\{ \varphi, \psi \} = i_1 \varphi + i_2 \psi.$$

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  • $\begingroup$ ------Thanks--- $\endgroup$ – Carlos I. Jan 19 '16 at 21:32
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In an abelian category $\oplus $ is the biproduct, that is $A\oplus B=A\times B$. If $\varphi:A\to B $ and $\psi :A\to C $ are two morphisms then there is a unique morphism $\delta :A\to B\times C=B\oplus C$ st $p\circ \delta=\varphi $ and $q\circ \delta=\psi$ , where $p $ and $q $ are the first and the second projection. Now $\{\varphi,\psi\} $ is $\delta $.

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  • $\begingroup$ ------Thanks--- – $\endgroup$ – Carlos I. Jan 19 '16 at 21:32

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