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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeJan 11th 2011
    • (edited Jan 11th 2011)

    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

    ProC(F,G)=lim dcolim cHom C(Fc,Gd) Pro-C(F,G) = lim_d colim_c Hom_C(Fc,Gd)

    to representatives ([s c,d]) d([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 dc dd\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 ?

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJan 11th 2011
    • (edited Jan 11th 2011)

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

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeJan 11th 2011

    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.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJan 11th 2011

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

    The objects of propro-CC are diagrams F:DCF:D\to C where DD is a small cofiltered category. The set of morphisms between F:DCF:D\to C and G:ECG:E\to C is

    pro-C(F,G)=lim eEcolim dDC(Fd,Ge)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 sproC(F,G)s\in\mathrm{pro}C(F,G) is a thread whose each component is a germ:
    s=(germ e(s)) eEs = (germ_e(s))_{e\in E} which can be more concretely written as ([s d e,e]) e([s_{d_e,e}])_e; thus [s d e,e]colim dDC(Fd,Ge)[s_{d_e,e}]\in colim_{d\in D} C(F d, G e) where s d e,eC(Fd e,Ge)s_{d_e,e}\in C(F d_e, G e) is some representative of the class; there is at least one d ed_e for each ee; if the domain EE is infinite, we seem to need an axiom of choice in general to find a function ed ee\mapsto d_e which will choose one representative in each class germ e(s)germ_e(s). Thus ss is given by the following data

    • function ed ee\mapsto d_e

    • correspondence es d e,eC(Fd e,Ge)e\mapsto s_{d_e,e}\in C(F d_e, G e)

    such that ([s d e,e]) e([s_{d_e,e}])_e is a thread, i.e. for any γ:ee\gamma: e\to e' we have an equality of classes (germs) [G(γ)s d e,e]=[s d e,e][G(\gamma)\circ s_{d_e,e}] = [s_{d_{e'},e'}]. This equality holds if there is a dd' and morphisms δ e:dd e\delta_e: d'\to d_e, δ e:dd e\delta_{e'}: d'\to d_{e'} such that G(γ)s d e,eFδ e=s d e,eFδ eG(\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 DD and the dual of CC as filtered domains). Now if we chose a different function ed˜ ee\mapsto\tilde{d}_e instead then, ([s d e,e]) e=([s d˜ e,e]) e([s_{d_e,e}])_e = ([s_{\tilde{d}_e,e}])_e, hence by the definition od classes, for every ee there is a dDd''\in D and morphisms σ e:dd e\sigma_e : d''\to d_e, σ˜ e:dd˜ e\tilde\sigma_e:d''\to \tilde{d}_e such that s d˜ e,eF(σ˜ e)=s d e,eF(σ e)s_{\tilde{d}_e,e}\circ F(\tilde\sigma_e) = s_{d_e,e}\circ F(\sigma_e).

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJan 11th 2011

    Now the composition !

    pro-C(F,G)=lim eEcolim dDC(Fd,Ge)pro\text{-}C(F,G) = lim_{e\in E} colim_{d\in D} C(F d, G e) pro-C(G,H)=lim bBcolim aEC(Ga,Hb)pro\text{-}C(G,H) = lim_{b\in B} colim_{a\in E} C(G a, H b)

    Seek for a map

    lim bBcolim aEC(Ga,Hb)×lim eEcolim dDC(Fd,Ge)lim bBcolim dDC(Fd,Hb)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.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJan 11th 2011

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

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJan 11th 2011
    • (edited Jan 11th 2011)

    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): sproC(F,G)s\in\mathrm{pro}C(F,G), tproC(G,H)t\in\mathrm{pro}C(G,H) is are given by functions ed ee\mapsto d_e, ba bb\mapsto a_b, and representatives ([s d e,e]) e([s_{d_e,e}])_e, ([t a b,b]) b([t_{a_b,b}])_b where s d e,eC(Fd e,e)s_{d_e,e}\in C(Fd_e,e) and t a b,bC(Ga b,Hb)t_{a_b,b}\in C(Ga_b,Hb). We need another function bd bb\mapsto d_b and ([(ts) d b,b) b([(t\circ s)_{d_b,b})_b and this is clearly the composition ba bd a bb\mapsto a_b\mapsto d_{a_b}. Furthermore,

    (ts) d a b,b=s d a b,a bt a b,b. (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 bt a b,b]) blim bBcolim dDC(Fd,Hb).([ 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 bbb\to b'), and then that the definition does not depend on various choices.