Are we not doing meetings anymore ? I did not get invites for last week or this week.
All posts by Arun Mathews
Remark about 5.1.1.13
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[2] to FLin[3] we need to pick 3 elements in FLin[3] so that they are related like they were in FLin[2]. The 3 elements we need to map are 0, 1, 2. If we map 0 in FLin[2] to 3 in FLin[3] then
f(0) = 3
In FLin[2] 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[3]. Rest is combinatorics.
Morphisms and homomorphisms
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 ?
Graph indexing category – 5.2.1.26
In section 5.2.1.21 author talks about graph-indexing category(GrIn) and symmetric graph-indexing category(SGrIn).
Ex. 5.2.1.26 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 ?
Question about Questionable Category (Remark 5.1.1.6 Book)
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.
Question about graph homomorphism
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 ?