Are there any books dealing with concrete categories besides "Joy of Cats"?

I find "Joy of Cats" hard to read and it is not always obvious that the concepts introduced are all that important or how they are important.

Please keep in mind that I'm specifically asking about concrete categories (those equipped with a faithful functor into another category, especially into $\mathsf{Set}$), not just categories.

As an example consider the questions: "How does a left-adjoint to the faithful functor into $\mathsf{Set}$ yield the concept of a subobject generated by a subset?" and "What property in my 'base' category is enough to ensure the existence of enough projectives in my concrete category?". These are (I'd say conceptually) questions about concrete categories, not mere categories.

  • $\begingroup$ How about Algebra by Mac Lane and Birkhoff? Or Categories for the Working Mathematician by Mac Lane? $\endgroup$ – lhf Dec 7 '16 at 10:16
  • $\begingroup$ I wish to add that examples in Abstract and Concrete Categories: The Joy of Cats are always without proofs, so good knowledge of those examples is required. Hence it is not possible to learn “concrete” branches of mathematics (abstract algebra, abstract analysis) employing category theory from the beginning. $\endgroup$ – beroal Feb 21 '18 at 17:02
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    $\begingroup$ @beroal Yeah, I think we are in desperate need of an actual introduction to concrete categories, not just a reference. $\endgroup$ – Stefan Perko Feb 21 '18 at 17:48
  • $\begingroup$ @StefanPerko I guess that we need to figure it out ourselves. $\endgroup$ – beroal Feb 22 '18 at 18:27

This book cannot exist.

A result due to Freyd states that any category with finite limits is concrete if and only if, for every object, its class of regular subobjects is, in fact, a set. This is a very mild condition to meet and thus the general theory of concrete categories is very close to being the theory of all abstract categories.

  • $\begingroup$ "Can be equipped with a functor to become a concrete category" is not the same as "is a concrete category"; or am I missing something? $\endgroup$ – Stefan Perko Dec 2 '18 at 10:41

You can see Kucera's paper here: http://www.sciencedirect.com/science/journal/00224049/1/4 on the topic that every category is a factorisation of a complex one. Hope it helps.


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