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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJan 5th 2013
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexSuppose we have a topos $H$ with a left-exact-reflective subcategory of codiscrete objects, with reflector $\sharp$, a reflective and coreflective subcategory of discrete objects with reflector $$$ and coreflector $\flat$, such that $\sharp\dashv\flat$ is an equivalence between the discrete and codiscrete objects, and this equivalence identifies $\sharp$ with $\flat$. Then we have an induced geometric morphism $p:H \to S$, where $S$ is the topos of discretes (or equivalently codiscretes), which is essential, connected, and local. The question is, what condition on $H$ and its modalities ensures that this geometric morphism is locally connected, i.e. that $p_!$ extends to an $S$-indexed left adjoint to $p^*$? The indexed version of $p^*$ is $p^*: S/X \to H/p^*X$, i.e. the inclusion of the objects in $H/A$ with discrete domain, for discrete $A$. So we need to know that this subcategory is reflective, and that the reflectors commute with pullback along maps between discrete objects. My first thought about this was, what if the reflector $$$ underlies a reflective subfibration or a stable factorization system. If the former, then for all $A\in H$ we have a reflective subcategory $D_A \subseteq H/A$ of "$A$-indexed discretes", the reflectors commuting with pullback. If the latter, then $D_A = M/A$ is the category of $M$-morphisms into $A$ for a factorization system $(E,M)$. And in the latter case, a morphism with discrete target is in $M$ precisely when its domain is discrete, so I think $D_A = M/A$ is precisely the subcategory we're interested in. If that's right, then enhancing $$$ to a stable factorization system may be just what we need. We've already noted that such an enhancement is useful for implying that $$$ preserves binary products---plus being easier to work with than escaping---so maybe this is really the way to go. But even though we decided this enhancement ought to be possible in the cases of interest, we never figured out whether there is a canonical or preferred way to do it.

   Suppose we have a topos $H$ with a left-exact-reflective subcategory of codiscrete objects, with reflector $\sharp$, a reflective and coreflective subcategory of discrete objects with reflector $$$ and coreflector $\flat$, such that $\sharp\dashv\flat$ is an equivalence between the discrete and codiscrete objects, and this equivalence identifies $\sharp$ with $\flat$.

   Then we have an induced geometric morphism $p:H \to S$, where $S$ is the topos of discretes (or equivalently codiscretes), which is essential, connected, and local. The question is, what condition on $H$ and its modalities ensures that this geometric morphism is locally connected, i.e. that $p_!$ extends to an $S$-indexed left adjoint to $p^*$?

   The indexed version of $p^*$ is $p^*: S/X \to H/p^*X$, i.e. the inclusion of the objects in $H/A$ with discrete domain, for discrete $A$. So we need to know that this subcategory is reflective, and that the reflectors commute with pullback along maps between discrete objects.

   My first thought about this was, what if the reflector $$$ underlies a reflective subfibration or a stable factorization system. If the former, then for all $A\in H$ we have a reflective subcategory $D_A \subseteq H/A$ of “$A$-indexed discretes”, the reflectors commuting with pullback. If the latter, then $D_A = M/A$ is the category of $M$-morphisms into $A$ for a factorization system $(E,M)$. And in the latter case, a morphism with discrete target is in $M$ precisely when its domain is discrete, so I think $D_A = M/A$ is precisely the subcategory we’re interested in.

   If that’s right, then enhancing $$$ to a stable factorization system may be just what we need. We’ve already noted that such an enhancement is useful for implying that $$$ preserves binary products—plus being easier to work with than escaping—so maybe this is really the way to go. But even though we decided this enhancement ought to be possible in the cases of interest, we never figured out whether there is a canonical or preferred way to do it.