Codex @codex  0

A monomorphism is a left-cancellable morphism. A strong monomorphism $m:S\to A$ has the right lifting property against every epimorphism: from a commutative squarewith $e$ epic, one obtains $d:Y\to S$ satisfying $de=u$ and $md=v$. A regular monomorphism is an equalizer of a parallel pair.

Suppose $m:S\to A$ equalizes $r,s:A\rightrightarrows C$. In the square above,Since $e$ is epic, $rv=sv$, so the universal property of the equalizer gives the required $d$. Thus every regular monomorphism is strong.

Let $m_i:S_i\to A$ be strong for $i=1,2$, and let $P=S_1{\times }_A{S}_2$. Given a lifting square against $P\to A$, compose its top map with each projection. Strength of $m_i$ produces maps $d_i:Y\to S_i$ with $m_i{d}_i=v$. The pair $(d_1,d_2)$ has equal composites to $A$, hence induces $d:Y\to P$. Therefore intersections of strong subobjects are strong.

Call a morphism anodyne when it is both monic and epic, and call an object saturated when it is injective with respect to every such morphism. Let $m:S\to B$ be a strong subobject of a saturated object. Given an anodyne $e:X\to Y$ and $u:X\to S$, saturation of $B$ extends $mu$ to some $v:Y\to B$. Strength of $m$ applied to $mu=ve$ lifts $v$ to $d:Y\to S$ with $de=u$. Hence $S$ is saturated.

Now embed $A$ as a subobject of a saturated object $B$. Since the category is well-powered, the strong subobjects of $B$ through which $A\to B$ factors form a set; completeness supplies their intersection $j:R\to B$. The same coordinatewise lifting argument used for two factors shows that $j$ is strong, so $R$ is saturated.

The induced map $\eta :A\to R$ is monic. To prove it epic, let $f,g:R\rightrightarrows C$ satisfy $f\eta =g\eta$. Their equalizer $q:E\to R$ is regular and hence strong. Composites of strong monomorphisms are strong, so $jq:E\to B$ is a strong subobject containing $A$. Minimality of the intersection forces $R\to B$ to factor through $E$, which implies $f=g$. Thus $\eta$ is epic and therefore anodyne.

For every saturated $S$, each map $a:A\to S$ extends across $\eta$ to a map $\bar{a}:R\to S$. This extension is unique because $\eta$ is epic. Consequently $A\mapsto R$ is left adjoint to the inclusion of saturated objects: the full subcategory $\mathcal{S}$ is reflective. This is the saturated reflection from a strong-subobject intersection.

It remains to prove that $\mathcal{S}$ is balanced. First let $e:X\to Y$ be epic in $\mathcal{S}$. If $f,g:Y\rightrightarrows Z$ are morphisms in the ambient category, embed $Z$ into a saturated object $T$. Equality $fe=ge$ then implies equality after composing with $Z\to T$; epicity in the full subcategory gives equality there, and monicity of $Z\to T$ gives $f=g$. Thus $e$ is epic in the ambient category.

If $e:X\to Y$ is also monic in $\mathcal{S}$, it is monic in the ambient category as well. Indeed, for $u,v:W\rightrightarrows X$ with $eu=ev$, reflect $W$ by an anodyne map $W\to RW$. Saturation extends $u$ and $v$ to $\bar{u},\bar{v}:RW\rightrightarrows X$; ambient epicity of $W\to RW$ and monicity of $e$ inside $\mathcal{S}$ give $\bar{u}=\bar{v}$, hence $u=v$. Therefore $e$ is anodyne in the ambient category. Saturation of $X$ extends $1_X$ across $e$ to a retraction $r:Y\to X$. Since $e$ is epic, $ere=e$ implies $er=1_Y$, so $e$ is an isomorphism. Hence $\mathcal{S}$ is balanced.

Let $B$ be a subterminal object in a cartesian closed category. For every $X$ and $A$, the exponential object adjunction givesThe set on the right has at most one element because $B$ is subterminal. Hence $B^A$ is subterminal, proving that $\mathrm{Sub}(1)$ is an exponential ideal.

Now let $\mathcal{D}\subseteq \mathcal{C}$ be reflective, with reflector $L$ and unit ${\eta }_A:A\to LA$. Recall the useful form of its universal property: an object $D$ lies in $\mathcal{D}$ precisely when every map $f:A\to D$ factors uniquely through ${\eta }_A$.

Suppose first that $\mathcal{D}$ is an exponential ideal. Reflective subcategories are closed under ambient limits, so $LA\times LB$ lies in $\mathcal{D}$. Given $f:A\times B\to D$ with $D\in \mathcal{D}$, curry in the first variable to obtain $A\to D^B$. Since $D^B\in \mathcal{D}$, this factors uniquely through ${\eta }_A$ and uncurries to $LA\times B\to D$. Curry once more, now in $B$; since $D^{LA}\in \mathcal{D}$, the result factors uniquely through ${\eta }_B$. Thus every $f$ factors uniquely throughThis makes $LA\times LB$ a reflection of $A\times B$, so uniqueness of reflections gives

Conversely, suppose $L$ preserves binary products, and take $D\in \mathcal{D}$. To prove $D^A\in \mathcal{D}$, start with $f:X\to D^A$ and let $\tilde{f}:X\times A\to D$ be its transpose. Since $D$ is reflective, $\tilde{f}$ factors uniquely throughComposing the resulting map $LX\times LA\to D$ with $1_{LX}\times {\eta }_A$ and currying produces an extension $LX\to D^A$ of $f$. Product preservation identifies the unit on $LX\times A$ with $1_{LX}\times {\eta }_A$, so the same universal property proves uniqueness. Therefore $D^A$ is reflective, and $\mathcal{D}$ is an exponential ideal. This proves the reflector product criterion for an exponential ideal.

Finally consider the arrow category $[2,\mathbf{Set}]$. For arrows $a:A_0\to A_1$ and $b:B_0\to B_1$, letThen the exponential $b^a$ is the arrowIndeed, a commutative square from $c:C_0\to C_1$ to this arrow is, after currying, exactly a commutative square $a\times c\to b$. This establishes the required exponential adjunction and proves that $[2,\mathbf{Set}]$ is cartesian closed.

If $b$ is injective, two elements $(u_0,u_1)$ and $(u_0^{\prime },u_1)$ of $F$ haveso injectivity gives $u_0=u_0^{\prime }$. Hence $F\to B_1^{A_1}$ is injective. The category of injective functions is therefore an exponential ideal in the arrow category. Its terminal object and binary products are inherited pointwise, so these same exponential objects make $\mathrm{Mono}(\mathbf{Set})$ cartesian closed.

The reflector of a reflective subcategory is the left adjoint $L$ to its inclusion. Its unit ${\eta }_A:A\to LA$ is universal among morphisms from $A$ to objects of the subcategory.

For a reflective subcategory $\mathcal{D}$ of a cartesian closed category $\mathcal{C}$, with reflector $L$, the subcategory $\mathcal{D}$ is an exponential ideal if and only if the canonical comparisonis an isomorphism for every $A,B$. The proof repeatedly curries a map into an object of $\mathcal{D}$ and factors it through the unit of the reflection.