# New meeting

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.

# 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).

How many functors are there of the form GrInSGrIn?

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.