creating a hash with elements from a different array of hashes in ruby

You can use Enumerable#each_with_object:

library.each_with_object({}) do |hash, result|
  result[hash['categories']] = hash['title']
end
# {
#   ["comedy", "fiction", "british"]=>"Hitchhiker's Guide",
#   ["morality", "fiction", "society", "british"]=>"Pride and Prejudice",
#   ["self improvement", "non-fiction", "mindfulness", "business"]=>"Search Inside Yourself",
#   ["non-fiction", "history", "founding fathers"]=>"Benjamin Franklin: An American Life",
#   ["play", "fiction", "drama"]=>"Glengarry Glen Ross"
# }

Here is another way to do it, which is less efficient than @AndreyDeineko's answer.

library.map{|h| h.values_at("categories", "title")}.to_h

Result:

{
  ["comedy", "fiction", "british"]=>"Hitchhiker's Guide", 
  ["morality", "fiction", "society", "british"]=>"Pride and Prejudice",
  ["self improvement", "non-fiction", "mindfulness", "business"]=>"Search Inside Yourself",
  ["non-fiction", "history", "founding fathers"]=>"Benjamin Franklin: An American Life",
  ["play", "fiction", "drama"]=>"Glengarry Glen Ross"
}

In case you wanted it to be distributed by the "category" attribute (which, of course, would be dependent on the order of the hash items in the array; I assume you meant later one has priority over an earlier one):

library.each_with_object({}) do
  |h, result| h["categories"].each_with_object(result) do
    |k, result| result[k] = h["title"]
  end
end

Result:

{
  "comedy"=>"Hitchhiker's Guide",
  "fiction"=>"Glengarry Glen Ross",
  "british"=>"Pride and Prejudice",
  "morality"=>"Pride and Prejudice",
  "society"=>"Pride and Prejudice",
  "self improvement"=>"Search Inside Yourself",
  "non-fiction"=>"Benjamin Franklin: An American Life", 
  "mindfulness"=>"Search Inside Yourself",
  "business"=>"Search Inside Yourself",
  "history"=>"Benjamin Franklin: An American Life",
  "founding fathers"=>"Benjamin Franklin: An American Life",
  "play"=>"Glengarry Glen Ross",
  "drama"=>"Glengarry Glen Ross"
}

If you wanted a list of each book in a category, and a book to appear in multiple categories, you can attach a default_proc when you initialize the Hash to set all unknown keys to contain an array as a value. Then you simply iterate each book in the library and add it's title to the list for every category it's in:

# Pass the category hash a default proc, which will be called
# whenever a key it doesn't have is accessed. So, when you
# encounter a brand new category, it'll already be set up with
# an array
category_hash = Hash.new { |hash, key| hash[key] = [] }
library.each do |book|
  book['categories'].each do |category|
    # since we passed the default, no worrying, an array will be
    # there and we can just add our value into it
    category_hash[category] << book['title']
  end
end

puts category_hash

Result:

{
  "comedy"=>["Hitchhiker's Guide"],
  "fiction"=>["Hitchhiker's Guide", "Pride and Prejudice", "Glengarry Glen Ross"],
  "british"=>["Hitchhiker's Guide", "Pride and Prejudice"],
  "morality"=>["Pride and Prejudice"],
  "society"=>["Pride and Prejudice"],
  "self improvement"=>["Search Inside Yourself"],
  "non-fiction"=>["Search Inside Yourself", "Benjamin Franklin: An American Life"],
  "mindfulness"=>["Search Inside Yourself"],
  "business"=>["Search Inside Yourself"],
  "history"=>["Benjamin Franklin: An American Life"],
  "founding fathers"=>["Benjamin Franklin: An American Life"],
  "play"=>["Glengarry Glen Ross"],
  "drama"=>["Glengarry Glen Ross"]
}

There's also each_with_object so you could modify this to

result = library.each.with_object(Hash.new { |hash, key| hash[key] = [] }) do |book, category_hash|
  book['categories'].each do |category|
    # since we passed the default, no worrying, an array will be
    # there and we can just add our value into it
    category_hash[category] << book['title']
  end
end
puts result

which would return the category_hash and, to save that last line of code:

puts(library.each.with_object(Hash.new { |hash, key| hash[key] = [] }) do |book, category_hash|
  book['categories'].each do |category|
    # since we passed the default, no worrying, an array will be
    # there and we can just add our value into it
    category_hash[category] << book['title']
  end
end)

This is a slight variant of other answers.

library = [
  {title: "Hitch Guide", by: "Doug", cats: ["comedy", "fiction", "british"]},
  {title: "P and P",     by: "Jane", cats: ["morality", "fiction", "british"]},
  {title: "Searchin'",   by: "Chad", cats: ["non-fiction", "morality", "gps"]},
  {title: "Crazy Ben",   by: "Walt", cats: ["non-fiction", "comedy", "dads"]}
]
  # => [{:title=>"Hitch Guide", :by=>"Doug", :cats=>["comedy", "fiction", "british"]},
  #     {:title=>"P and P", :by=>"Jane", :cats=>["morality", "fiction", "british"]},
  #     {:title=>"Searchin'", :by=>"Chad", :cats=>["non-fiction", "morality", "gps"]},
  #     {:title=>"Crazy Ben", :by=>"Walt", :cats=>["non-fiction", "comedy", "dads"]}]

library.each_with_object({}) do |g,h|
  title = g[:title]
  g[:cats].each { |c| (h[c] ||= []) << title }
end
  #=> {"comedy"=>["Hitch Guide", "Crazy Ben"],
  #    "fiction"=>["Hitch Guide", "P and P"],
  #    "british"=>["Hitch Guide", "P and P"],
  #    "morality"=>["P and P", "Searchin'"],
  #    "non-fiction"=>["Searchin'", "Crazy Ben"],
  #    "gps"=>["Searchin'"],
  #    "dads"=>["Crazy Ben"]}

This is how I would normally do it and I think it's the simplest possible way, although the each_with_object trick seen above is neat too.

new_hash = {}

library.each do |hash| 
  hash["categories"].each do |category|
    new_hash[category] ||= []
    new_hash[category] << hash["title"]
  end
end