Isomorphism-invariance and categorical properties

Question

I was studying category theory when a question came to my mind.

It is often argued that, given a mathematical object, its categorical properties are exactly those which are isomorphism-invariant. What does this mean? I'd like to make this statement more precise, and not being enough competent I'm asking a question about.

I suppose this somehow presupposes the presence of a categorical language (which, roughly speaking, has as sorts objects and morphisms and as operations the composition, the identity, domain and codomain; in that, it applies equally to all categories) and to call a categorical property (or universal property) a one-variable predicate in that language. At this point, I suppose the statement in bold could be rephrased saying that, given two objects of a fixed category, they are isomorphic if and only if they satisfy the same categorical properties.

Is this the way? How could one make it formal or precise?

EDIT Thanks to present answers, we've found out that isomorphic objects share the same categorical properties, whereas the converse may not be true. Is there a study about the categories in which this holds, i.e., what are the categories in which two objects having the same categorical properties are isomorphic? Is Set one of those?

Answer 1

The "if and only if" in your final interpretation is not correct (as shown by Qiaochu), but otherwise it looks good. The correct statement is as follows.

Let $P$ be a "natural" property in the language of category theory. Assume that $A \cong A'$ and that $A$ satisfies $P$. Then also $A'$ satisfies $P$. There is also an obvious version with more arguments and parameters.

The proof will be a simple induction once we define our properties exactly, and in particular those which are "natural". Most importantly, $A=B$ is not natural in this regard, and clearly this property is not isomorphism invariant. We should in general disallow any statements which assert that two objects are equal. Notice that no definition in category theory appearing in practice does something like that. We simply don't ask questions like "There are exactly three objects which admit a morphism to $A$".

We can, however, assert that two parallel(!) morphisms are equal. Equality of non- parallel morphisms is a type mismatch and hence not natural.

For a typical example, notice that "$A$ is projective" or "$A$ has all copowers" are natural properties, and indeed are isomorphism-invariant.

Answer 2

This is essentially exactly the topic of Freyd's 1976 paper Properties Invariant Within Equivalence Types of Categories (though as the other answers explain, isomorphism-invariance isn't quite the appropriate property to consider). The paper begins thus:

All of us know that any "mathematically relevant" property on categories is invariant within equivalence types of categories. Furthermore, we all know that any "mathematically relevant" property on objects and maps is preserved and reflected by equivalence functors. An obvious problem arises: How can we conveniently characterize such properties?

In the paper, Freyd introduces a diagrammatic language for categories and proves that a property of categories is invariant under equivalence if and only if it is a property expressible in the diagrammatic language.

Answer 3

This is not correct. Categorical properties are also invariant under all automorphisms of the category.

For a trivial example, consider the category $\{ \bullet, \bullet \}$ consisting of two objects and no non-identity morphisms; these two objects are not isomorphic but share all categorical properties, because they are swapped by the non-identity automorphism of the category itself.

For a more "natural" example, consider the category of $\mathbb{Z}$-graded vector spaces $V = \bigoplus V_i$. This category has a shift automorphism $V_i \mapsto V_{i+1}$, and all categorical properties, e.g. being finite-dimensional, are invariant under this automorphism, even though it generally will not send objects to isomorphic objects.