Hi,
Let $\mathcal{C}$ be a small category with finite limits and finite colimits.
The Ind-completion $Ind(\mathcal{C})$ is a locally finitely presentable category which is the free cocompletion of $\mathcal{C}$ under directed colimits. That is, there is a full embedding $I : \mathcal{C} \hookrightarrow Ind(\mathcal{C})$ and for any functor $H : \mathcal{C} \to \mathcal{D}$ where $\mathcal{D}$ has directed colimits, there is an essentially unique directed colimit preserving extension $H^* : Ind(\mathcal{C}) \to \mathcal{D}$ with $H^* \circ I = H$.
The latter uses the fact that $\mathcal{C}$ has finite colimits. We can also construct the Pro-completion $Pro(\mathcal{C})$. Since $\mathcal{C}$ has finite limits it arises as the free completion of $\mathcal{C}$ under inverse limits and is dual to the locally finitely presentable category $Ind(\mathcal{C}^{op})$.
Then there is an adjunction $F \dashv U$ where $U : Pro(\mathcal{C}) \to Ind(\mathcal{C})$ is the unique inverse limit preserving extension of $\mathcal{C} \hookrightarrow Ind(\mathcal{C})$, likewise $F$ uniquely extends $\mathcal{C} \hookrightarrow Pro(\mathcal{C})$. For example if $\mathcal{C} = \mathsf{FinSet}$ then essentially $U : \mathsf{Stone} \to \mathsf{Set}$ is the usual forgetful functor and $F$ is the Stone-Cech compactification.
Questions:
- Is it known whether $U$ and $F$ are always faithful and preserve both epis and monos?
- Are any other general preservation properties known?
- Aside from "Stone Spaces", what is a good source for the connection between Pro and Ind completions?
Any help much appreciated.
Here are some examples where $U$ and $F$ are both faithful and both preserve epis and monos.
- If $\mathcal{C}$ is complete or cocomplete then it is essentially a complete lattice (since $C$ is small), so $Ind(\mathcal{C}) \cong Pro(\mathcal{C}) \cong C$ and $U$, $F$ define an equivalence.
- If $\mathcal{C} = \mathsf{FinSet}$ then $U : \mathsf{Stone} \to \mathsf{Set}$ is the forgetful functor, whose left adjoint is the Stone-Cech compactification.
- If $\mathcal{C} = \mathsf{FinBA}$ then $U : \mathsf{CABA} \to \mathsf{BA}$ is the faithful forgetful functor, whose left adjoint could be described as the canonical extension of a boolean algebra.
- If $\mathcal{C} = \mathsf{FinDL}$ then $U : \mathsf{Priestley} \to \mathsf{Poset}$ is the forgetful functor, whose left adjoint is the ordered Stone-Cech compactification.
- If $\mathcal{C} = \mathsf{FinPoset}$ then $U : \mathsf{DADL} \to \mathsf{DL}$ is the faithful forgetful functor where $\mathsf{DADL}$ is the category of doubly algebraic distributive lattices with complete lattice morphisms. Its left adjoint could be described as the canonical extension of a distributive lattice.
- If $\mathcal{C} = \mathsf{FinJSL}_\bot$ (finite join semilattices with bottom) then $U : \mathsf{JSL}_\bot^{op} \to \mathsf{JSL}_\bot$ sends semilattices to their ideal completion and morphisms to the right adjoint of their extension.
- If $\mathcal{C} = \mathsf{FDVect}(\mathbb{F})$ (finite dimensional vector spaces) then $U : \mathsf{Vect}(\mathbb{F})^{op} \to \mathsf{Vect}(\mathbb{F})$ is the usual functor sending a vector space to its dual space.
- If $\mathcal{C} = \mathsf{FinAb}$ (finite abelian groups) then $U : \mathsf{Ab}(\mathsf{Stone}) \to \mathsf{TorAb}$ is the forgetful functor from the Stone topological abelian groups to abelian groups with torsion.
Admittedly I haven't checked all the details, but I'd also be interested in other such examples.
