Is every abstract category a concrete category of structures?

In conceptual mathematics by Lawvere and Schanuel, it says:

There are many more categories than just those given by abstract types of structure; however, those can be construed as full subcategories of the latter, so that the notion of map does not change.

Given a type of structure, one has structures of that type. Morphisms are structure preserving maps from one structure to another; composition is defined as in the category of sets.

Is every category isomorphic to a subcategory of a category of structures of a given type? Or how to interpret the above statement from the book?

Is there a precise notion of "type of structure" (the notion given in the book surely isn't the only one) such that every category is isomorphic to a concrete category of structures.

No, it's not the case.

Let $C$ be a category of "structures" (whatever it means). To it is naturally associated a so-called forgetful functor that strips away all the structure and leaves only the underlying set.

This is thus a functor $U: C\to \text{Set}$. This functor has the remarkable property that it is faithful: if $f, g: A\to B$ have the same image under $U$ ($U(f)= U(g)$) then they must be equal. That's because for "structures", moprhisms are given by "structure"-respcting functions, so when you strip out the structure, well the function stays the same.

Any subcategory $D$ of $C$ also has a faithful functor $U_{\mid D} : D\to \text{Set}$.

However, as it turns , there are categories without a faithful functor to $\text{Set}$, these categories can therefore not be subcategories of a category of structures (for a sensible notion of structure)

there are categories without a faithful functor to Set

There are in fact many such categories. https://arxiv.org/abs/1704.00303

• Very interesting ! +1 – Max Sep 23 '17 at 9:22