The discussion focuses on the verification of the coproduct universal property in the context of free products of groups, specifically G * H. Participants emphasize that the construction is designed to fulfill the universal property, which is more about notation than mathematics. They highlight the importance of understanding the direction of arrows in category theory and recommend consulting nLab for deeper insights. The conversation also touches on the construction of unique morphisms and the role of amalgamated free products.
PREREQUISITESMathematicians, particularly those specializing in group theory and category theory, as well as students seeking to understand the complexities of coproducts and free products in algebra.
You basically just write it down carefully. It is a matter of notation rather than a matter of mathematics. It is constructed to fulfill the universal property.Heidi said:
Why don't you try what I have said? Write down all morphisms that you have, name the group and the index set, and note what must be shown. I see no sense in rewriting the definition. So in order to communicate, we need a common language, not just M*N. Have you read the nLab link?Heidi said:
Ok, this simply means ##H=\{1\}.## The elements of ##G=G_1\ast G_2## are words over the alphabet of elements from ##G_1## and ##G_2.## Say we have ##abcxyacbza\in G## with ##a,b,c\in G_1## and ##x,y,z\in G_2.## Then we have a homomorphism ##f_1\, : \,G_1\longrightarrow Q## that maps ##a,b,c,## say onto ##\overline{a},\overline{b},\overline{c}## and the same for ##f_2\, : \,G_2\longrightarrow Q## that maps ##x,y,z## onto ##x',y',z'.## The dashed group homomorphism ##f## maps then ##abcxyacbza## onto ##f_1(a)f_1(b)f_1(c)f_2(x)f_2(y)f_1(a)f_1(c)f_1(b)f_2(z)f_1(a)= \overline{a}\overline{b}\overline{c}x'y'\overline{a}\overline{c}\overline{b}z'\overline{a}.##Heidi said: