Proving Non-Concreteness of a Category

Question

So, since I've just accidentally stumbled onto the topic of non-concrete (inconcrete?) categories, I've been looking around for some simple examples that did not even make use of such things as elements, and on the blog Mathematics Prelims, I found a rather neat example:

There are, of course, categories which are not concrete. In the last post we saw a very simple category with three objects. These objects aren’t sets, they’re just “things” that we decided to label A, B, and C. The morphisms between these objects aren’t functions in any sense of the word: they don’t associate inputs with outputs, there aren’t even any “inputs” or “outputs” to speak of! Our morphisms are literally just arrows that start at one object and end at another. (For this reason some people simply refer to the morphisms of a category as arrows.)

Well, only thing that bothers me now is my head goes around in circles as I try to figure out how to definitively prove that you can not construct a faithful functor from this category to Set.

Anyone willing to help satisfy my urge for completeness on this matter?

Look forward to your responses.

Answer 1

There is an important terminological discrepancy here: some authors use the term "concrete" for what others refer to as "concretizable" (see e.g. nLab versus Freyd). After a certain amount of exposure, this discrepancy is fairly benign, but initially it can be confusing. Contra the original version of this answer, I'll follow the nLab language.

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Despite the name, a concrete category isn't just a category - it's a category together with a particular faithful functor into Sets. A concretizable category, meanwhile, is a category such that there exists such a functor (but we don't pick out a specific one).

So the linked post is correct in claiming that the category in question is not a concrete category - for the silly reason that it's just a category, as opposed to a category + a specific faithful functor into Sets. On the other hand, it's easy to show that it is concretizable.

• It's worth observing that all non-concretizable categories are pretty complicated (and in particular, all small categories are concretizable).

That said, the post linked is still slightly imprecise when it comes to defining concrete categories:

[C]oncrete categories [...] are categories where the objects are sets, usually with some additional structure (group structure, a topology, etc.), and the morphisms are well-defined functions between those sets that preserve the structure.

(Emphasis mine.) The bolded "are" isn't really accurate; rather, a concrete category is intuitively a category together with an "interpretation" of the objects and morphisms as sets and functions. But the objects aren't required to literally be sets, they just need to correspond to sets in some explicit way (namely, via a specific choice of faithful functor).

Answer 2

Call this category $\mathbf{C}$. Then, there are lots of functors from $\mathbf{C}$ to $\mathbf{Set}$, and they are all faithful because $\mathbf{C}$ is a poset.

Answer 3

The naive homotopy category of pointed topological spaces is not conrete. Here is how we construct it: $\textbf{Top}_*$ is the category of pointed topological spaces and continuous basepoint preserving functions between them.

Let $\text{NH}(\textbf{Top}_*)$, the naive homotopy category of pointed topological spaces be the category with the same objects as $\textbf{Top}_*$ but the hom sets $\text{Hom}_{\text{NH}(\textbf{Top})}((A,a),(B,b))$ are pointed homotopy classes of maps $(A,a) \rightarrow (B,b)$. I.e two maps are in the same class if there is a pointed homotopy between them.

Basic homotopy theory tells us that the usual composition of maps gives a well defined composition of maps on homotopy classes. I.e if we let $[f]$ be the set of pointed homotopy classes then the operation $[f] \circ[g] = [f \circ g]$ is well defined.

The proof that $\text{NH}(\textbf{Top}_*)$ is not concrete is complicated but you can read it here

Answer 4

There are plenty of examples of things that are naturally categories, but do not have a canonical way to be embedded in Set (here I am taking concrete not as concretizable, but equipped with a specific faithful functor to Set).

• Take a graph and consider the free category generated by this graph, i.e. the objects are vertices of the graph, and the arrows are paths from one vertex to another in your graph. This is a perfectly fine category, which is not concrete. This leads to my favorite example: starting from the graph of the metro in New York (or anywhere you want). The objects are literally metro stations, and the morphisms are metro trips... This is both very natural and as far from sets as I can imagine

• In type theory, we look at the syntactic category associated to a theory, where the objects are contexts and the morphisms are substitutions. They don't look like sets

• When you define a monoid as a one object category, the only object that you consider has nothing to do with sets a priori

So I am aware that these examples may be concretizable, but I like the fact that they appear really naturally and are not at all formulated with sets.