Compatible family in a presheaf

Question

Let C be a category. Let c be an object in C. One definition I know for a compatible family on c in a presheaf F is that it is a natural transformation from R to F where R is a sieve on c. Another definition which I know, is a family $(x_f|f\in R)$ where R is a sieve on c such that for all appropriate g, we have $x_{fg}=F(g)(x_f)$. I was trying to look at $(x_f|f\in R)$ as a set which I came up with the question that what these really are as a set, where they exist. Since if we look at them as natural transformations, if they be equal, we get that the sieves which they have been defined on them, are equal, I guess as a set they should be defined in a way that also we can get this.

Answer 1

Since this is a question about why two definitions are essentially the same thing, it is important to be precise about what these definitions are precisely. Throughout we fix a small category $\mathcal{C}$. If we want to be really precise we also have to adopt the convention that hom-sets are disjoint (i.e. domain and codomain are part of the data of an arrow).

Definition. A sieve on an object $C$ in $\mathcal{C}$ is a set $R$ of arrows in $\mathcal{C}$ with codomain $C$. Such that if $f: D \to C$ is an arrow in $R$ and $g: E \to D$ is any arrow in $\mathcal{C}$, then $fg \in R$.

Recall that a subobject $G$ of a presheaf $F$ in $\mathbf{Set}^{\mathcal{C}^\text{op}}$ consists of a subset $G(C) \subseteq F(C)$ for each object $C$ in $\mathcal{C}$. Such that for any arrow $f: C \to D$ and $x \in G(D)$ we have that $F(f)(x) \in G(C)$. Of course, technically speaking anything isomorphic to such $G$ would be a subobject, but any subobject can be canonically represented as just described. So it is easier to think of them in this way.

Let $y: \mathcal{C} \to \mathbf{Set}^{\mathcal{C}^\text{op}}$ denote the Yoneda-embedding and write $y_C$ and $y_f$ for the embedding of $C$ and $f$ respectively. The point is that a sieve on $C$ is precisely the data of a subobject of $y_C$ in $\mathbf{Set}^{\mathcal{C}^\text{op}}$. That is, given a sieve $R$ on $C$, we can define a presheaf $G_R$ by setting $$ G_R(D) = \{ f \in R : \operatorname{dom}(f) = D \}. $$ Conversely a subobject $G$ of $y_C$ yields a sieve $R_G$ by setting $$ R_G = \bigcup_{D \text{ an object in } \mathcal{C}} G(D). $$ One easily checks that these operations are indeed well-defined. In particular, we see that the requirement $f \in R \implies fg \in R$ is precisely saying that $f \in G_R(D) \implies y_C(g)(f) \in G_R(E)$ (for any $g: E \to D$). And the same when we start with $G$ and build $R_G$.

So a sieve is a set of arrows in $\mathcal{C}$, and what I denoted by $G_R$ is a subobject in $\mathbf{Set}^{\mathcal{C}^\text{op}}$ and is thus in particular a functor (however you want to encode that). Of course this is only a question about encoding these things, because the above shows that they are essentially the same thing.

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The link to compatible families is that a compatible family in some presheaf $F$ for a sieve $R$ is the same thing as a natural transformation $G_R \to F$. That is, given a compatible family $(x_f : f \in R)$ we can define a natural transformation $\xi: G_R \to F$ by letting $\xi_D: G_R(D) \to F(D)$ be given by $$ \xi_D(f) = x_f. $$ Conversely, given a natural transformation $\gamma: G_R \to F$ we get a compatible family $(y_f : f \in R)$ by setting $$ y_f = \gamma_{\operatorname{dom}(f)}(f). $$

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So yes, it only makes sense for two natural transformations to be the same, if their domains are the same. So if we view compatible families $(x_f : f \in R)$ and $(x'_f : f \in R')$ as natural transformations $\xi: G_R \to F$ and $\xi': G_{R'} \to F$, then they can only be the same if $G_R = G_{R'}$, i.e. $R = R'$. This makes sense, because two compatible families are the same if they are pointwise equal, and how would we do that if their indexing sets are different?

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Edit. Part of the confusion seemed to come from the notation $(x_f \mid f \in R)$. As explained in the comments: this just means we have a set indexed by $R$. Put differently, it is a function $R \to \bigcup_{C \text{ an object in } \mathcal{C}} F(C)$ sending $f$ to $x_f$. Note the similarity with the natural transformation $\xi$ above, which comes down to breaking this function up for each object of $\mathcal{C}$.