Are we not doing meetings anymore ? I did not get invites for last week or this week.
The author is asking about Homomorphism of FLin. In section 4.4.4 he talks about morphism of orders. If s1 <= s2 then f(s1) <= f(s2). Also, order is reflexive (s <= s). So when we have a morphism from FLin to FLin we need to pick 3 elements in FLin so that they are related like they were in FLin. The 3 elements we need to map are 0, 1, 2. If we map 0 in FLin to 3 in FLin then
f(0) = 3
In FLin 0 <= 1. So f(0) <= f(1). If f(0) is 3 then f(1) and f(2) has to be 3 for f(0) <= f(1) and f(0) <= f(2) and f(1) <= f(2). Basically we need to find 3 non decreasing elements in FLin. Rest is combinatorics.
In section 5.1.1 where category is defined the author says
B. for every pair x, y ∈ Ob(C), a set HomC(x,y)∈Set; it is called the hom-set from x to y; its elements are called morphisms from x to y;2
Footnote 2 says
The reason for the notation Hom and the word hom-set is that morphisms are often called homomorphisms, e.g., in group theory.
But morphisms and homomorphisms are different, right ? Morphism can be anything, homomorphisms are more restrictive. Is that correct ? For eg: there can be morphisms between 2 groups that are not homomorphisms, right ?
In section 188.8.131.52 author talks about graph-indexing category(GrIn) and symmetric graph-indexing category(SGrIn).
Ex. 184.108.40.206 asks
How many functors are there of the form GrIn → SGrIn?
I did not understand the answer and explanation given in the textbook. Can anyone please explain ?
In the remark author says:
A. there is a function U:Ob(Q)→Ob(C),
B. for all a,b∈Ob(Q), we have an injection U:HomQ(a,b)↪HomC(U(a),U(b)),
Why is the function and injection both called U ? Shouldn’t the injection be called something other than U ? This is more a question of naming but I just wanted to confirm.
According to the definition:
A graph (V, A, src, tgt) involves two sets and two functions. For two graphs to be comparable, their two sets and their two functions should be appropriately comparable. Let G = (V, A, src, tgt) and G′ = (V′, A′, src′, tgt′) be graphs. A graph homomorphism f from G to G′, denoted f : G → G′, consists of two functions f0: V → V′ and f1: A → A′ …the rest…
So this means that not all elements of A’ need to have a mapping from A and not all elements of V’ need to have a mapping from V. For example if G has m vertices and n edges and G’ has m'(>1) vertices and n'(>1) edges we could have all vertices of G go to one vertex of G'(say a) and all edges of G go to an edge from a to a(assuming that exists) in G’. That would be a valid homomorphism. Am I understanding that correctly ?