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Concerning your first question: d(h) = d∘h + h∘d (by definition), so Hinich's definition is the standard one. For the second question, having d(h) = id defines a contractible chain complex, since d(h) = d∘h + h∘d = id.


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The short answer to your main question is "no". If $A_*$, $B_*$, and $C_*$ are chain complexes, having $C_q = A_q \oplus B_q$ for each $q$ in some sense doesn't guarantee that $C_* = A_* \oplus B_*$ as chain complexes. The relevant maps between $C_q$ and $A_q \oplus B_q$ need to also respect the boundary maps in the chain complexes in order for ...


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The universal property of limits is the following: Given a set $\{A_i\}_{i\in\Bbb N}$ of objects, and a diagram $$\cdots \to A_3\to A_2\to A_1$$ (where we say $f_i:A_{i+1}\to A_i$), the limit of this diagram is an object $A$ and a set of morphisms $\{g_i:A\to A_i\}_{i\in \Bbb N}$ such that $g_i=f_i\circ g_{i+1}$ for all $i\in \Bbb N$ For any other object $...


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The splitting lemma does not apply in this situation. Also for $L \circ T$ to be bijective $L$ must be surjective and $T$ injective. The following statement is true for all composition of maps. $L \circ T$ is bijective iff $T$ is injective and $L|_{im T} $ is bijective. When looking at linear maps this translates to: $L \circ T$ is bijective iff $T$ is ...


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