I am looking for reference on cup and cap product without invoking acyclic model theorem. To me, acyclic model theorem is very strange phenomena though I could understand it but I do not see direct construction.
$\textbf{Q:}$ Is there a reference on cup and cap product construction without invoking acyclic model theorem(relative and $C(X)\otimes_Z C(Y)\cong C(X\times Y)$ as quasi isomorphism)? I would like to see a direct (computable) construction which will demonstrate non-commuativity of cup, associativity of both cap and cup whenever they are well defined. I am having trouble to see "obviously" $u\cup v=(-1)^{deg(v)deg(u)}v\cup u$ as well.(Note here I should not have written in this way as $u\in H^i(X), v\in H^j(X)$ but I have identified $H^{i+j}(X\times Y)=H^{i+j}(Y\times X)$ in the image. This is indicating diagram is commutative up to a sign.) Most of time, the book proves this by acyclic model via producing homotopy to a chain map with a sign.
The cup product is graded commutative in homology, but not on the chain level. Maps witnessing higher non-commutativity in a coherent way are known as $i$-cup products, and were introduced by N. Steenrod in this paper. Computations there are very explicit. The fundamental result for (usual) cup products is that if $a$ and $b$ are cochains in degree $p$ and $q$, there is a cochain $a\smile_1 b$, called the 1-cup product of $a$ with $b$, so that
$$d(a\smile_1 b) -da\smile_1 b-(-1)^p a\smile_1 b= (-1)^{p+q+1}[a,b]$$
where the right hand side is the graded commutator. This MO post contains more information on these operations.