Closed Monoidal Categories and the Internal Hom - Examples

The evaluation map is $\mathrm{ev}:C^B\times B\to C$, $(\varphi ,b)\mapsto \varphi (b)$. For example, $\mathrm{ev}(xy,0)=x$ and $\mathrm{ev}(xy,1)=y$.

Let $A=\{\ast \}$. A map $A\times B\to C$ is just a function $B\to C$, so it is exactly one of the four displayed elements of $C^B$. Hence maps $A\times B\to C$ biject with maps $A\to C^B$ by sending the unique point of $A$ to the corresponding function.

This is the exponential universal property in the concrete case, matching Set is cartesian closed.

Currying in the $C$-variable sends $(a,b)$ to the function $c\mapsto a+b+c$. Currying again in the $B$-variable sends $a$ to the function $b\mapsto (c\mapsto a+b+c)$.

Uncurrying the result reverses those two assignments: from $a$ recover the map $b\mapsto (c\mapsto a+b+c)$, then evaluate at $(b,c)$ to recover $a+b+c=f(a,b,c)$.

So the twice-curried and twice-uncurried maps return the original $f$, exactly as Currying and uncurrying are mutually inverse predicts.

The evaluation morphism is the group homomorphism $\mathrm{ev}:{\mathrm{Hom}}_{\mathbf{Z}}(B,C){\otimes }_{\mathbf{Z}}B\to C$, $\varphi \otimes b\mapsto \varphi (b)$. It is bilinear, so it is well defined.

By Hom-tensor adjunction: ${\mathrm{Hom}}_R(M{\otimes }_{R}N,P)\cong {\mathrm{Hom}}_R(M,{\mathrm{Hom}}_R(N,P))$, for every abelian group $A$ there is a natural isomorphism $\mathrm{Hom}(A\otimes B,C)\cong \mathrm{Hom}(A,\mathrm{Hom}(B,C))$. This is exactly the transposition property from The internal hom and its evaluation morphism.

Therefore ${\mathrm{Hom}}_{\mathbf{Z}}(B,C)$ is the internal hom in $\mathbf{A}\mathbf{b}$.

To give a functor $F:\mathbf{I}\to \mathcal{E}$ is to choose the two objects $F(0),F(1)$ and the image $F(u):F(0)\to F(1)$. Thus the objects of the exponential ${\mathcal{E}}^{\mathbf{I}}$ are precisely the arrows of $\mathcal{E}$.

A morphism between two such functors is a natural transformation, so it is exactly a commutative square in $\mathcal{E}$. Therefore ${\mathcal{E}}^{\mathbf{I}}$ is the arrow category of $\mathcal{E}$.

This is the concrete exponential object described by The category of small categories is cartesian closed: functors $\mathcal{C}\times \mathbf{I}\to \mathcal{E}$ curry to functors $\mathcal{C}\to {\mathcal{E}}^{\mathbf{I}}$.

The dependent product ${\mathrm{\Pi }}_{f}E\to Y$ has fiber over $0$ equal to $E_a\times E_b=\{(u,w),(v,w)\}$ and fiber over $1$ equal to $E_c=\{z\}$. So ${\mathrm{\Pi }}_{f}E$ has three elements in total, two over $0$ and one over $1$.

If $q:B\to Y$ is the singleton fiber over $1$, then $f^{\ast }q\to X$ is the singleton fiber over $c$, and a map $f^{\ast }q\to p$ over $X$ is exactly a choice of an element of $E_c$, namely $z$. This matches maps $q\to {\mathrm{\Pi }}_{f}E$ over $Y$, which must choose the unique element over $1$.

This concrete fiber computation realizes the pullback/right-adjoint pattern guaranteed by Set is locally cartesian closed, using the slice formalism of Slice categories, composition, and pullback along a morphism.

On the walking arrow, the arrows into $0$ are only $1_0$, so its sieves are $\mathrm{\varnothing }$ and $\{1_0\}$. The arrows into $1$ are $u$ and $1_1$, and downward closure forces any sieve containing $1_1$ also to contain $u$, giving exactly the three displayed sieves on $1$. Pullback of sieves makes $\mathrm{\Omega }$ a presheaf, and the maximal sieves define the natural truth morphism above.

Consider the representable presheaf $y(1)$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding) and the subpresheaf $S\subset y(1)$ defined by $S(1)=\mathrm{\varnothing }$ and $S(0)=\{u\}$. Its characteristic map $\chi :y(1)\to \mathrm{\Omega }$ sends the unique element $u\in y(1)(0)$ to the maximal sieve $\{1_0\}$, and sends $1_1\in y(1)(1)$ to the sieve $\{u\}$.

For a general subpresheaf $T\subseteq P$, the displayed formula defines a sieve and is natural under restriction. Moreover ${\chi }_{T,c}(x)$ is maximal exactly when $x\in T(c)$: the forward implication tests $1_c$, and the reverse follows from closure of the subpresheaf under restriction. Thus pulling back $\mathrm{true}$ along ${\chi }_T$ recovers $T$, and this condition uniquely determines ${\chi }_T$. By [L1], $\mathrm{\Omega }$ is a subobject classifier.

For the concrete $S\subset y(1)$ of step 2.1, that pullback includes $u$ at object $0$ because ${\chi }_0(u)=\{1_0\}$ is maximal, and excludes $1_1$ at object $1$ because ${\chi }_1(1_1)=\{u\}$ is not maximal. Hence the general classifier calculation gives exactly the displayed example.