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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeJan 11th 2011
   • (edited Jan 11th 2011)
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   Author: zskoda
   Format: MarkdownItexPeople around me use a concrete definition of morphisms between pro-objects via the representatives. I will sketch this later this afternoon, but it is well-known and obvious. But if one passes from $$ Pro-C(F,G) = lim_d colim_c Hom_C(Fc,Gd) $$ to representatives $([s_{c,d}])_d$, which are threads of germs then one notices that such a representative if written explicitly is in fact involving a choice of a function $d\mapsto c_d$, hence to describe the maps as threads of germs one seems to need an axiom of choice. So suppose we do not have an axiom of choice. Do we have then two non-equivalent categories of pro-objects ?

   People around me use a concrete definition of morphisms between pro-objects via the representatives. I will sketch this later this afternoon, but it is well-known and obvious. But if one passes from

   $$Pro-C(F,G) = lim_d colim_c Hom_C(Fc,Gd)$$

   to representatives $([s_{c,d}])_d$, which are threads of germs then one notices that such a representative if written explicitly is in fact involving a choice of a function $d\mapsto c_d$, hence to describe the maps as threads of germs one seems to need an axiom of choice. So suppose we do not have an axiom of choice. Do we have then two non-equivalent categories of pro-objects ?

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJan 11th 2011
   • (edited Jan 11th 2011)
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   Author: zskoda
   Format: MarkdownItexIn fact I do not like the $Pro$-$C$ or $pro-C$ kind of notation from [[pro-object]]. I mean I see no point in dash here, and ${Pro}C$ or $Pro(C)$ find better.

   In fact I do not like the $Pro$-$C$ or $pro-C$ kind of notation from pro-object. I mean I see no point in dash here, and ${Pro}C$ or $Pro(C)$ find better.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJan 11th 2011
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   Author: zskoda
   Format: MarkdownItexI wrote some references into [[pro-object]] and struggle with writing up the derivation of the elementary definition, using axiom of choice. Sorry, wait a bit until it is a correct version.

   I wrote some references into pro-object and struggle with writing up the derivation of the elementary definition, using axiom of choice. Sorry, wait a bit until it is a correct version.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeJan 11th 2011
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   Author: zskoda
   Format: MarkdownItexHere we are (this is now extended central part of [[pro-object]] where I have written the explicit description): The objects of $pro$-$C$ are diagrams $F:D\to C$ where $D$ is a [[small category|small]] [[filtered category|cofiltered]] category. The set of morphisms between $F:D\to C$ and $G:E\to C$ is \[pro\text{-}C(F,G) = lim_{e\in E} colim_{d\in D} C(F d, G e)\] The limit and colimit is taken in the category of sets; we know that cofiltered limits there are "threads" and filtered colimits are germs (classes of equivalences). Thus a representative of $s\in\mathrm{pro}C(F,G)$ is a thread whose each component is a germ: $s = (germ_e(s))_{e\in E}$ which can be more concretely written as $([s_{d_e,e}])_e$; thus $[s_{d_e,e}]\in colim_{d\in D} C(F d, G e)$ where $s_{d_e,e}\in C(F d_e, G e)$ is some representative of the class; there is at least one $d_e$ for each $e$; if the domain $E$ is infinite, we seem to need an axiom of choice in general to find a function $e\mapsto d_e$ which will choose one representative in each class $germ_e(s)$. Thus $s$ is given by the following data * function $e\mapsto d_e$ * correspondence $e\mapsto s_{d_e,e}\in C(F d_e, G e)$ such that $([s_{d_e,e}])_e$ is a thread, i.e. for any $\gamma: e\to e'$ we have an equality of classes (germs) $[G(\gamma)\circ s_{d_e,e}] = [s_{d_{e'},e'}]$. This equality holds if there is a $d'$ and morphisms $\delta_e: d'\to d_e$, $\delta_{e'}: d'\to d_{e'}$ such that $G(\gamma)\circ s_{d_e,e}\circ F\delta_e = s_{d_{e'},e'}\circ F\delta_{e'}$. (Usually in fact people consider the dual of $D$ and the dual of $C$ as filtered domains). Now if we chose a different function $e\mapsto\tilde{d}_e$ instead then, $([s_{d_e,e}])_e = ([s_{\tilde{d}_e,e}])_e$, hence by the definition od classes, for every $e$ there is a $d''\in D$ and morphisms $\sigma_e : d''\to d_e$, $\tilde\sigma_e:d''\to \tilde{d}_e$ such that $s_{\tilde{d}_e,e}\circ F(\tilde\sigma_e) = s_{d_e,e}\circ F(\sigma_e)$.

   Here we are (this is now extended central part of pro-object where I have written the explicit description):

   The objects of $pro$-$C$ are diagrams $F:D\to C$ where $D$ is a small cofiltered category. The set of morphisms between $F:D\to C$ and $G:E\to C$ is

   $$pro\text{-}C(F,G) = lim_{e\in E} colim_{d\in D} C(F d, G e)$$

   The limit and colimit is taken in the category of sets; we know that cofiltered limits there are “threads” and filtered colimits are germs (classes of equivalences). Thus a representative of $s\in\mathrm{pro}C(F,G)$ is a thread whose each component is a germ:
   $s = (germ_e(s))_{e\in E}$ which can be more concretely written as $([s_{d_e,e}])_e$; thus $[s_{d_e,e}]\in colim_{d\in D} C(F d, G e)$ where $s_{d_e,e}\in C(F d_e, G e)$ is some representative of the class; there is at least one $d_e$ for each $e$; if the domain $E$ is infinite, we seem to need an axiom of choice in general to find a function $e\mapsto d_e$ which will choose one representative in each class $germ_e(s)$. Thus $s$ is given by the following data

   • function $e\mapsto d_e$

   • correspondence $e\mapsto s_{d_e,e}\in C(F d_e, G e)$

   such that $([s_{d_e,e}])_e$ is a thread, i.e. for any $\gamma: e\to e'$ we have an equality of classes (germs) $[G(\gamma)\circ s_{d_e,e}] = [s_{d_{e'},e'}]$. This equality holds if there is a $d'$ and morphisms $\delta_e: d'\to d_e$, $\delta_{e'}: d'\to d_{e'}$ such that $G(\gamma)\circ s_{d_e,e}\circ F\delta_e = s_{d_{e'},e'}\circ F\delta_{e'}$. (Usually in fact people consider the dual of $D$ and the dual of $C$ as filtered domains). Now if we chose a different function $e\mapsto\tilde{d}_e$ instead then, $([s_{d_e,e}])_e = ([s_{\tilde{d}_e,e}])_e$, hence by the definition od classes, for every $e$ there is a $d''\in D$ and morphisms $\sigma_e : d''\to d_e$, $\tilde\sigma_e:d''\to \tilde{d}_e$ such that $s_{\tilde{d}_e,e}\circ F(\tilde\sigma_e) = s_{d_e,e}\circ F(\sigma_e)$.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJan 11th 2011
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   Author: zskoda
   Format: MarkdownItexNow the composition ! \[pro\text{-}C(F,G) = lim_{e\in E} colim_{d\in D} C(F d, G e)\] \[pro\text{-}C(G,H) = lim_{b\in B} colim_{a\in E} C(G a, H b)\] Seek for a map \[lim_{b\in B} colim_{a\in E} C(G a, H b) \times lim_{e\in E} colim_{d\in D} C(F d, G e)\to lim_{b\in B} colim_{d\in D} C(F d, Hb)\] It is easy to write it down in terms of germs and threads, but now I look to see it from general nonsense point of view (I mean first principles), before writing it up into the [[pro-object|entry]].

   Now the composition !

   $$pro\text{-}C(F,G) = lim_{e\in E} colim_{d\in D} C(F d, G e)$$ $$pro\text{-}C(G,H) = lim_{b\in B} colim_{a\in E} C(G a, H b)$$

   Seek for a map

   $$lim_{b\in B} colim_{a\in E} C(G a, H b) \times lim_{e\in E} colim_{d\in D} C(F d, G e)\to lim_{b\in B} colim_{d\in D} C(F d, Hb)$$

   It is easy to write it down in terms of germs and threads, but now I look to see it from general nonsense point of view (I mean first principles), before writing it up into the entry.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJan 11th 2011
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   Author: Mike Shulman
   Format: MarkdownItexIn the absence of AC I would just use an [[entire relation]] instead of a function.

   In the absence of AC I would just use an entire relation instead of a function.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeJan 11th 2011
   • (edited Jan 11th 2011)
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   Author: zskoda
   Format: MarkdownItexThanks, Mike, I will think about your advice! Continuing old story ... just for the record. In terms of threads and germs (just for the record): $s\in\mathrm{pro}C(F,G)$, $t\in\mathrm{pro}C(G,H)$ is are given by functions $e\mapsto d_e$, $b\mapsto a_b$, and representatives $([s_{d_e,e}])_e$, $([t_{a_b,b}])_b$ where $s_{d_e,e}\in C(Fd_e,e)$ and $t_{a_b,b}\in C(Ga_b,Hb)$. We need another function $b\mapsto d_b$ and $([(t\circ s)_{d_b,b})_b$ and this is clearly the composition $b\mapsto a_b\mapsto d_{a_b}$. Furthermore, $$ (t\circ s)_{d_{a_b},b} = s_{d_{a_b},a_b}\circ t_{a_b,b}. $$ Now one needs to show that this is well defined and that indeed it defines a thread of germs, more precisely a well defined element $$([ s_{d_{a_b},a_b}\circ t_{a_b,b}])_b \in lim_{b\in B} colim_{d\in D} C(F d, Hb).$$ There are two things to check: first that one indeed has a thread (compatibility relation for all morphisms $b\to b'$), and then that the definition does not depend on various choices.

   Thanks, Mike, I will think about your advice!

   Continuing old story … just for the record.

   In terms of threads and germs (just for the record): $s\in\mathrm{pro}C(F,G)$, $t\in\mathrm{pro}C(G,H)$ is are given by functions $e\mapsto d_e$, $b\mapsto a_b$, and representatives $([s_{d_e,e}])_e$, $([t_{a_b,b}])_b$ where $s_{d_e,e}\in C(Fd_e,e)$ and $t_{a_b,b}\in C(Ga_b,Hb)$. We need another function $b\mapsto d_b$ and $([(t\circ s)_{d_b,b})_b$ and this is clearly the composition $b\mapsto a_b\mapsto d_{a_b}$. Furthermore,

   $$(t\circ s)_{d_{a_b},b} = s_{d_{a_b},a_b}\circ t_{a_b,b}.$$

   Now one needs to show that this is well defined and that indeed it defines a thread of germs, more precisely a well defined element

   $$([ s_{d_{a_b},a_b}\circ t_{a_b,b}])_b \in lim_{b\in B} colim_{d\in D} C(F d, Hb).$$

   There are two things to check: first that one indeed has a thread (compatibility relation for all morphisms $b\to b'$), and then that the definition does not depend on various choices.