Why do these equivalent categories seem to behave differently?

Question

Write $\{1\}$ for the terminal category, and let $\Omega$ denote a category with two distinct (but isomorphic objects), call them $1_\Omega$ and $0_\Omega$, and a total of four arrows; two identities, an arrow $1_\Omega \leftarrow 0_\Omega$, and another arrow $0_\Omega \leftarrow 1_\Omega$.

Write $j : \Omega \leftarrow \{1\}$ for the functor given by $j(1)=1_\Omega.$

Then intuitively, the pair $(\Omega,j)$ should be able to classify the full subcategories of any other category. Explicitly, given a category $\mathbf{C}$ and a subcategory $\mathbf{S},$ there is a functor $P_\mathbf{S} : \Omega \leftarrow \mathbf{C}$ given as follows:

• $P_\mathbf{S}(X)=1$ iff $X \in \mathbf{S}$
• $P_\mathbf{S}(X)=0$ iff $X \notin \mathbf{S}$

The arrows go the only place they can go.

Furthermore, if we restrict to full subcategories, then this process should be reversible.

But this makes no sense!

In particular, the category $\Omega$ is equivalent to the category $\{1\}$, and hence should have no advantages over $\{1\}.$ Yet we cannot classify full subcategories using the arrow $\{1\} \leftarrow \{1\}$.

What on earth is going on here? Why do these equivalent categories seem to behave differently?

Answer 1

Your confusion is the following: you have two categories $1$ and $\Omega$ which are equivalent, but not isomorphic. You note that for any category $C$ there is a unique functor $C \to 1$ but that functors $C \to \Omega$ seem to correspond to full subcategories.

There is a tight analogy between categories and topological spaces (so tight it can be realized by a functor, the geometric realization of the nerve) that goes like this:

• Categories are like spaces.
• Functors are like maps between spaces.
• Isomorphisms of categories are like homeomorphisms of spaces.
• Natural transformations are like homotopies.
• Functor categories are like spaces of maps between spaces.
• Equivalences of categories are like homotopy equivalences of spaces.

In particular, two spaces (say the point and the interval $[0, 1]$, which are quite analogous to the terminal category and $\Omega$) can be homotopy equivalent, but not homeomorphic, and hence the set of maps from some space $X$ into those two spaces can look very different. However, it's always true that if two spaces are homotopy equivalent, then the spaces of maps from $X$ to those two spaces are also homotopy equivalent: for example, the space of maps from a space $X$ into $[0, 1]$ is always contractible.

The corresponding statement for categories is that if two categories are equivalent, then the categories of functors (and natural transformations) from any other category $C$ to those categories are equivalent. In particular, we conclude that the category of functors $C \to \Omega$ must in fact be equivalent to the terminal category.

So, what are the morphisms in this category anyway? Suppose $F, G : C \to \Omega$ are two functors. What is a natural transformation between them? Well, it's a collection of maps $\eta_c : F(c) \to G(c)$ such that... but wait. In $\Omega$ there's a unique morphism between any two objects. Hence all of the $\eta_c$ are in fact uniquely determined, and automatically satisfy the necessary compatibility condition to define a natural transformation. Moreover, every such natural transformation is invertible because its inverse is also uniquely determined!

At this point let me introduce the following useful terminology: a category is contractible if there is a unique morphism between any two objects. This condition is equivalent to being equivalent to the terminal category. What we've written down is more or less a proof that if $\Omega$ is a contractible category, then so is any functor category $[C, \Omega]$.

Answer 2

Isomorphic categories are interchangeable in a very strong sense. Basically, whenever you have an isomorphism $A\cong B$, you can interchange $A$ for $B$ in any categorical situation. To make this more precise, just use first order logic. Let $T$ be the first order theory of categories; A $T$-model is the same as a category (add small where needed to avoid annoyances). An isomorphism of categories is the same an a $T$-isomorphism. $T$-isomorphic structures are interchangeable within the world of $T$-models and $T$-questions (i.e., problems formulated in the language of $T$ and using the axioms of $T$). This is why isomorphic categories are interchangeable within category theory; the symmetries of category theory include all permutations of isomorphic categories.

Equivalence of categories is something of a different nature. Equivalent categories can't be interchanged within category theory. The correct motto is that working inside category $A$ or inside an equivalent category $A'$ does not (essentially) matter. That is certainly true, but the emphasis here is on the word 'inside'. From the outside (but still within category theory) two equivalent categories may certainly look very different. A way to look at it is to think of a category as the category of objects modeling something. Equivalent categories offer concrete models which may be very different to what must essentially be the same thing. More precisely (and ignoring set theoretic issues again), define a 'thing' to be an equivalence class of categories (the equivalence relation is equivalence of categories). Then every category in a thing $t$, i.e., a representative of $t$, offers concrete models of $t$. Any category equivalent to it offers other models of the same $t$. It does not matter in which representative you work since you are actually interested in $t$ (the axioms defining the concrete category you actually choose for the models are just a way to turn the vague 'thing' $t$ into actual mathematical entities). However, when you can look around and see all categories and ask all sorts of categorical questions, it is not only $t$ that you suddenly see. That is why internally equivalent categories are the same, but externally they may be very different.

Answer 3

Ah, but are you really classifying subcategories? What if you have two subcategories $S$ and $S'$. This gives us the Functors $P_S$ and $P_{ S'}$, but are these functors really different? Well, we can define a natural isomorphism $!$ that for each object in $C$ is assigned the corresponding isomorphism in $\Omega$. So $P_S$ and $P_{S'}$ are basically the same Functor, because they send everything to basically the same object. It would be like distinguishing things by marking some with on pencil and others with another, technically different by essentially the same as marking them all with the same pencil.

In general, comparing Functors for equality is something called evil, precisely because it is broken under equivalence. Evil concepts generally are no different in practice from their nonevil counterparts, isomorphism in this case. See http://ncatlab.org/nlab/show/principle+of+equivalence#in_category_theory

All essentially concepts are preserved by equivalences. Just make sure not to differentiate essentially the same things, like isomorphic Functors.