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At pag 15.

Theorem 1.6 Every category C with a set of arrows is isomorphic to one in which the objects are sets and the arrows are functions.

The following Remark 1.7 is:

"This shows us what is wrong with the naive notion of a 'concrete' category of sets and functions: while not every category has special sets and functions as its objects and arrows, every category is isomorphic to such a one. Thus the only special properties such categories can posses are the ones that are categorically irrelevant, such as features of the objects that do not affect the arrows in any way"

Theorem 1.6 is the generalization to categories of Cayley theorem for groups (every group is isomorphic to a group of permutations), which is useful because the isomorphism allows to study the group of permutations and obtain information on every other group.

  1. My Problem is the sentence in bold. One one side isomorphic in a way means that two things are the same, so how can one have no categorical interest while on the other side of the iso there's something that is of categorical interest (in the sense of the properties they possess)?
  2. Also it seems to me like Awodey is saying that studying concrete categories is categorically useless, because of the generalized Cayley, while for groups the group of permutations I'd say acquires importance thanks to Cayley's theorem. I am not understanding why for categories happens the opposite than what happens for groups.
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I guess that what Awodey is telling that the previous given definition of concrete category (i.e. a category having structured sets as objects and structure preserving functions as arrows) is not a good one because every category is isomorphic to a such category (so every category should be concrete, at least for this definition). If all categories are concrete it's meaningless introduce this definition of concrete category because it doesn't really characterize anything new.

Indeed after that Awodey introduce a new notion of concrete category

"A better attempt to capture what is intended by the rather vague idea of a “concrete” category is that arbitrary arrows f : C → D are completely determined by their com- posites with arrows x : T → C from some “test object” T , in the sense that f x = gx for all such x implies f = g. This amounts to considering a particular representation of the category, determined by T . A category is then said to be “concrete” when this condition holds for T a ”terminal object”, in the sense of section 2.2 below; but there are also good reasons for considering other objects T , as we shall see in the next chapter".

This definition of concrete category characterize a special kind of categories (not all categories have a test object $T$ as above, less have $T$ as terminal object). So this is a useful definition because introduce something new.

Edit: Just because we can see every group as a permutation group doesn't mean that we should always do that, sometimes the details of permutations can hide what the really important data we need to prove a result, making harder our task at hand. This argument applies to category theory too. But that's not all. When someone starts studying category theory one of the thing that seems to distinguish arrows by function is that the second one are completely characterized by their actions on elements, meaning that two functions $f,g \colon X \to Y$ are equal iff for every element $x \in X$ the equality $fx=gx$ holds.

Later one finds out that this isn't actually true: indeed given a category $\mathcal C$ and two arrows $f,g \in \mathcal C(X,Y)$ they are equal iff for every generalized element $x \in \mathcal C(A,X)$ the equality $f \circ x = g \circ x$ holds.

This observation allows one to translate many arguments used to proving facts in concrete categories such $\mathbf{Set}$, $\mathbf{Grp}$, $\mathbf{Top}$ and so on, in other categories. This trick is used in homological algebra (i.e. Abelian category theory) for doing the diagram chasing in non set-function categories, but also in categorical logic for building the internal logic of a category.

Hope this help.

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  • $\begingroup$ I see what you mean. The "old" concrete category "definition" added nothing new, so it was "not interesting". I'd say point 1 is now clarified (I'd upvote if I could). On the other side, I am still a bit confused about the comparison with the Cayley theorem for groups and how useful it is to have something isomorphic to everything. $\endgroup$ – Dirich May 8 '13 at 18:38
  • $\begingroup$ @Dirich what do you mean by how useful it is to have something isomorphic to everything? $\endgroup$ – Giorgio Mossa May 8 '13 at 21:10
  • $\begingroup$ Since every group is isomorphic to a permutation group, you can study permutation groups and know things about every group. So it is useful to study permutation groups. This is what I meant. $\endgroup$ – Dirich May 9 '13 at 7:34
  • $\begingroup$ Ok, now I get it. I'll try to add something to the answer addressing this point. $\endgroup$ – Giorgio Mossa May 9 '13 at 9:36
  • $\begingroup$ The edit clarified the second point. Thank you! $\endgroup$ – Dirich May 9 '13 at 12:06

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