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Forte Agent 4.1.1088 Crack.Q:
Why is the coproduct of two free associative algebras (in the category of vector spaces) equal to their tensor product?
In the following three lines,
the first two clearly show the coproduct is equal to the tensor product, and the third clearly shows the coproduct is not equal to the tensor product. However, this comes from the definition of the tensor product.
If not, how does one get from the first two to the third?
$(V\oplus W)\otimes X \simeq V\otimes X\oplus W\otimes X$
$(V\otimes W)\oplus X \simeq V\oplus(W\otimes X)$
$(V\oplus W)\otimes X=V\oplus(W\otimes X)$
A:
For the first equality, the definition of tensor product is:
$$(V \oplus W) \otimes X = V \otimes X \oplus W \otimes X$$
And this is just:
$$(V \oplus W) \otimes X = (V \otimes X \oplus W \otimes X)$$
This equality is valid in any category with the canonical projection to $V$ and $W$ defined.
If you want to use a distributive lattice instead, you have to define a coproduct in the category. This is done on page 29 of
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