[Math] Adjunction between Pro and Ind Completions

ct.category-theory

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.

  1. 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.
  2. If $\mathcal{C} = \mathsf{FinSet}$ then $U : \mathsf{Stone} \to \mathsf{Set}$ is the forgetful functor, whose left adjoint is the Stone-Cech compactification.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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.

Best Answer

No a answere but a specification and description of the issue (too long for a comment).

The free cocompletion property of $Ind(\mathcal{C})$ dont use the hypothesis "$\mathcal{C}$ has finite colimits". ANyway $Ind(\mathcal{C})$ is equivalent to the full subcategory $P_1(\mathcal{C})\subset \mathcal{C}^>$ of presheves that are coimits of a small, filtred diagram of representable. In this way the inclusion $i$ corresponds to the yoneda inclusion $h_-: \mathcal{C}\to P_1(\mathcal{C})$ and it preserves limit, and finite colimits. The inclusion $P_1(\mathcal{C})\subset \mathcal{C}^>$ create (small) filter colimits (if $P=\varinjlim_{i\in I}P_i$ is a filtrant colimit with $O_i\in P_1(P_1(\mathcal{C})$ then combining the comma categories $P_1(\mathcal{C})\downarrow P_i$ we make a filtrant (small) diagram of $P_1(\mathcal{C})\downarrow P_i$

Is $F: \mathcal{C}\to \mathcal{D}$ where $\mathcal{D}$ as filter colimits we have a (iso)unique extentions $F': P_1(\mathcal{C})\to \mathcal{D}$ with $F'(P):=\varinjlim_{\ (X, x)\in \mathcal{C}\downarrow P} F(X)$, if $P$ has the Ind-representation $(X_i)_{i\in I}$ i.e. $P=\varinjlim_i h_{X_i}$ then the diagram of the $h_{X_i}$ is a final diagram on the comma category $\mathcal{C}\downarrow P$ then $F'(P):=\varinjlim_i F(X_i)$

If $\mathcal{C}$ has finite colimits $P_1(\mathcal{C})\cong Cart(\mathcal{C}^{op}, Set)$ the latter is the the category cartesians presheaves i.e. that maps finite colimits of $\mathcal{C}$ to finite limits in $Set$, of course the embedding $Cart(\mathcal{C}^{op}, Set)\subset \mathcal{C}^>$ create limits .

All above as a dual version for $Proj(\mathcal{C}):=(Ind(\mathcal{C}^{op}))^{op}$, it is a free completion of $\mathcal{C}$, and if $\mathcal{C}$ has finite limits $Proj(\mathcal{C})$ is equivalent to $Cart(\mathcal{C}, Set)^{op}$ (dual to the category of copresheaves that preserving finite limits) it has (small) colimits and we have the embedding $\iota:=(h^-)^{op}: \mathcal{C}\to Cart(\mathcal{C}, Set)^{op} $.

Now the inclusion $h_-: \mathcal{C}\to Cart(\mathcal{C}^{op}, Set)$ and $\iota: \mathcal{C}\to Cart(\mathcal{C}, Set)^{op}$ induce for the universal properties of these two completions the functors $U: Cart(\mathcal{C}, Set)^{op}\to Cart(\mathcal{C}^{op}, Set)$ with $U(Q)=\varprojlim_{\ h^Y\to Q} h_Y$, and $F: Cart(\mathcal{C}^{op}, Set)\to Cart(\mathcal{C}, Set)^{op}$ with

$F(P)=\varprojlim_{\ h_X\to P} h^X$ (the limit is in $\mathcal{C}^{<}$ the copresheaves category)

we have the natural isomorphisms:

$Cart(\mathcal{C}, Set)^{op}(F(P), Q)=Cart(\mathcal{C}, Set)(Q, F(P))=$

$\mathcal{C}^{<}(\varinjlim_{\ h^Y\to Q}h^Y, \varprojlim_{\ h_X\to P}h^X)=$

$\varprojlim_{\ h_X\to P} \varprojlim_{h^Y\to Q}\mathcal{C}^<(h^Y, h^X)\cong$

$\varprojlim_{\ h_X\to P} \varprojlim_{h^Y\to Q}\mathcal{C}^>(h_X, h_Y)\cong$

$\mathcal{C}^>(\varinjlim_{\ h_X\to P} h_X, \varprojlim_{h^Y\to Q}h_Y) \cong$

$Cart(\mathcal{C}^{op}, Set)(P, U(Q))$

then $U$ is a adjoint to $F$.

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