Please let me know if the following statements betray a misunderstanding as I build towards my question.
A category $\mathbf{C}$ is a class $\text{Obj}_C$ of objects and a set $\text{Hom}_C$ of morphisms. For each morphism $f$ there are objects $\text{dom}(f)$ and $\text{cod}(f)$. There is an identity morphism for every object, there is a morphism for every composition of morphisms, and composition is associtative.
A morphism contains three distinct types of data:
- reference to the domain object
- reference to the codomain object
- structural data about the relationship between the domain and codomain.
The structural data may be trivial (a morphism connects domain $A$ and codomain $B$) or fundamental to what the category is trying to describe (linear maps in $\mathbf{Vect}$).
The following is an example of a category in which the morphisms tell us nothing about the objects. Let $\mathbf{Dumb}$ be a category where $\text{Obj}=\{S_5,\mathbf{R}^3\}$ and $\text{Hom}=\{f,1_{S_5},1_{\mathbf{R}^3}\}$. Here, $f$ is a morphism whose domain is the symmetric group $S_5$ and codomain is the vector space $\mathbf{R}^3$. It contains no information other than this.
Question: What does a category's morphisms tell us about its objects? Would it be fair to say that in a "useful" category, the morphisms tell us what structure we're interested in preserving amongst a family of objects? For example, linearity is what makes vector spaces interesting, so we choose linear maps as the morphisms in $\mathbf{Vect}$. Then, the actually "arrangement" of morphisms amongst the objects give us additional information about their structure.
From this perspective, morphisms fulfill two roles
- identify objects' "structure" we're interested in preserving (looking within)
- provide insight into this structure via the connections they form (looking without)
Is this assessment accurate? I feel like I am punching my way through fog trying to precisely understand basic categorical notions. Any and all insight appreciated!
Best Answer
The real question here is what properties the language of category theory captures.
A statement in the (finitary) language of category theory is one formed from propositions of the form $f = g$ where $f, g : X \to Y$ using the propositional operations of $\lor, \land, \top, \bot, \neg, \implies$, together with quantifiers either over all objects (eg $\forall X$, $\exists X$ where $X$ is an object variable) or over all morphisms between two given objects (eg $\forall f : X \to Y$ or $\exists f : X \to Y$ where $X, Y$ are objects).
Two things can be shown about statements in the language of category theory. Suppose given a statement $\phi(X_1, ..., X_n, f_1, ..., f_m)$ in the language of category theory with object variables $X_1, ..., X_n$ and free function variables $f_i : X_{d_i} \to X_{c_i}$ for $1 \leq i \leq m$.
Now consider a category and two different variable assignments - one assignment $X_1 \mapsto W_1, ..., X_n \mapsto W_n$ where each $W_i$ is an object, and $f_1 \mapsto g_1, ..., f_m \mapsto g_m$ where $g_i : W_{d_i} \to W_{c_i}$ for all $i$, and another assignment $X_1 \mapsto Y_1, ..., X_n \mapsto Y_n$ and $f_1 \mapsto h_1, ..., f_m \mapsto h_m$, where $Y_i$ is an object for all $i$ and $h_j : W_{d_j} \to W_{c_j}$ for all $j$. Suppose there are isomorphisms $k_1 : W_1 \to Y_1, ..., k_n : W_n \to Y_n$ such that for all $i$, $h_i = k_{c_i} \circ f_i \circ k_{d_i}^{-1}$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m) \iff \phi(Y_1, ..., Y_n, h_1, ..., h_m)$. This is known as "isomorphism invariance of truth".
Now consider a functor $F : C \to D$ which is fully faithful and essentially surjective, and a variable assignment in $C$ of the form $X_i \mapsto W_i$, $f_j \mapsto g_j$ as above. We assume that $\phi$ has no free variables other than the $X_i$ and $f_i$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m)$ iff $\phi(F(W_1), ..., F(W_n), F(g_1), ..., F(g_m))$. This is known as "equivalence invariance of truth".
Both of the above can be proved by induction on formulas.
The two above statements can be generalised to the infinitary language of category theory, which allows quantification over external sets, though some care must be taken in settings without the axiom of choice.
So the language of category can only discuss properties which are invariant up to isomorphism and up to equivalence of categories. No finer-grained discrimination is possible.
For example, consider the category of sets. It follows from isomorphism invariance that the language of category cannot distinguish two sets which have the same cardinality. The language of category cannot distinguish a monic $f : S \to T$ from a subset $S \subseteq T$.
For the category of topological spaces with morphisms the continuous maps quotiented by homotopy, the language of category theory cannot distinguish between homotopy equivalent spaces.
So the key when using category theory is to find the right amount of data for morphisms to carry based on the underlying subject one is studying and the particular context.