Motivation: In May's Concise Course in Algebraic Topology, the colimit is brought up to talk about van Kampen. Here's the diagram (from wikipedia) for ease of notation:
He notes that $\phi_{X}, \phi_{Y}$ are the initial such morphisms into the colimit $L$. In most cases, it's clear that this means something like "an injection." Although,...
Questions:
- Because the colimit is an object satisfying this diagram, how can we force $\phi_{X}$ and $\phi_{Y}$ to be initial if we are not a priori considering the colimit object $L$? It doesn't seem like we can take initial morphisms in a hom-set where the destination object is not fixed.
- Similar to the first part, when people talk about the colimit of some objects, they are implicitly specifying two "injection" mappings (in this case $\phi_{X}$ and $\phi_{Y}$); how do we pick these?
As an example for the second, if we consider the diagram for the product in the category of Groups and make the "projections" just zero mappings, then we will not obtain the same product as if we make them the "standard" projections --- I'm not entirely sure if we will get a product at all in that case.