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I am reading Awodey (Category Theory, 1st edition), p 175, and I have difficulties to understand the paragraph about the subobject classifier of $\mathbf{Sets}^{\mathbf{C^{op}}}$.

First let me quote the paragraph. Awodey is trying to define a subobject classifier $1 \rightarrow \Omega$ by using sieves:

Let $$ \Omega(C) = \{ S\subseteq > \mathbf{C}_1 \mid S \text{ is a sieve > on } C \} $$

and given $h:D \rightarrow C$ let

$$ h^* : \Omega(C)\rightarrow\Omega(D) > $$

be defined by

$$ h^*(S)=\{g:\cdot \rightarrow D \mid > h \circ g \in S \} $$

This clearly defines a preasheaf $\Omega : \mathbf{C^{op}} \rightarrow > \mathbf{Sets}$, with a distinguished point,

$$ t:1\rightarrow\Omega $$

namely, at each $C$, the "total sieve"

$$ t_C = \{ f : \cdot \rightarrow C\} > $$

We claim that $t:1\rightarrow \Omega$ so defined is a subobject classifier for $\mathbf{Sets}^{\mathbf{C^{op}}}$. Indeed, given any object $E$ and a subobject $U \rightarrow E$, define $u > : E \rightarrow \Omega$ at any object $C \in \mathbf{C}$ by:

$$ u_C(e) = \{f:D\rightarrow C \mid > f^*(e) \in U(D) \rightarrowtail E(D) > \} $$

for any $e \in E(C)$. That is, $u_C(e)$ is the sieve of arrows into $C$ that take $e \in E(C)$ back into the subobject $U$.

At this point I am very troubled by the notation $f^*(e)$.

If I assume it is a pullback notation, being the pullback functor of $f$ applied on the element $e$, I end up with this diagram:

$$ \begin{matrix} &\xrightarrow{} & 1 \\[1ex] \downarrow \rlap{\scriptstyle{f^*(e)}} & & \downarrow \rlap{\scriptstyle{e}} \\[1ex] &\xrightarrow{f} & E(C) \\[1ex] \end{matrix} $$

that seems to me totally crazy because $f$ is supposed to be of the type $f:D\rightarrow C$.

Could somebody show me what I missed?

Note: If possible, could you avoid using adjoints in your answers? Just because I have not yet reached the chapter when it is defined and explained... (other notions are ok)

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2 Answers

What's going on is that, since $E$ is a contravariant functor to $\mathbf{Sets}$, by definition $f:D \to C$ induces a map of sets $f^*:E(C) \to E(D)$. Then $u_C(e)$ consists of the arrows $f$ such that $f^*(e)$ actually lands in the subobject $U(D)$ of $E(D)$. You can then check that this is a sieve for each $e$, and so on.

It might be helpful to let $E$ be a representable functor, say $E = \mathbf{C}(-,X)$. Then $e$ is a map $C \to X$, $U(D)$ is a set of maps $D \to X$, and $u_C(e)$ is the set of maps $D \to C$ such that the composition $D \to C \stackrel{e}{\to} X$ is in the set $U(D)$.

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Thanks a lot! I appreciate very much your example. It shows why the notation $f^*$ has been used, acting by precomposition on the map $e$. –  Almeo Maus May 6 '13 at 16:04
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Isn't $f^{*}(e)$ just the action of the functor $E$ on morphisms, i.e., it could be written instead as $E(f)(e)$? Perhaps Steve can confirm in person.

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Steve says the first edition has a rather unfortunate number of typos. –  Andrej Bauer May 6 '13 at 15:12
    
Thank you very much for your answer! With your definition $f^∗=E(f)$, it works. I have also the 2nd edition in book version (the 1st ed. is my portative electronic edition), and the notations are the same for this paragraph ($f^∗$). –  Almeo Maus May 6 '13 at 16:05
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