The discussion centers on the construction of the free product of groups and its verification of the coproduct universal property. Participants explore the theoretical underpinnings, notation, and implications of this construction within category theory.
Participants express differing levels of understanding regarding the construction's verification of the universal property, indicating that multiple competing views remain. The discussion does not reach a consensus on the clarity or sufficiency of the explanations provided.
Some participants highlight the complexity of the definitions and the need for careful notation, suggesting that assumptions about the construction may not be universally understood. The discussion also touches on the potential confusion arising from different types of universal constructions in category theory.
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: