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In looking at pro-categories and prohomotopy, I find statements in the literature to the effect that every constant pro-object is cosmall, and then (Christensen and Isaksen):Every object of every pro-category is κ-cosmall relative to all pro-maps for some κ. The proof that they give seems to me a bit like reinventing the wheel. Isn’t this something like the dual of the arguments used in looking at locally FP categories and Gabriel-Ulmer duality? Their result is used in a lot of the papers on pro-homotopy theories as then these are (very nearly) fibrantly generated.
I need this for my monograph on profinite homotopy, but we have nothing on cosmall objects and the consequences of the cosmall object argument in the nLab, and intend putting a version of it there afterwards.
Does anyone have thoughts on how to present this in the Lab. (I will have to give more (tedious) detail in the monograph as I do not have LFP categories explained anyway.) I also feel that some of the gory detail given more or less categorical folklore, but have not been able to track down enough to be able to pin that down. (It is almost in SGA4 which is online.) I am hindered by not having access to a library as I work from home. (Oh for universal open access!!!!!)
Every object in $Ind (\mathcal{C})$ is small (relative to all morphisms in $Ind (\mathcal{C})$) (for some cardinal depending on the object). This is because the class of small objects is closed under small colimits (in any category). But $Pro (\mathcal{C}) = Ind (\mathcal{C}^{op})^{op}$, so I suppose your statement follows by duality.
Thanks … and representables are small…. is that the point? I had written down that but thought that if it was so simple why had C & I written such a lot on it! I feared I might be missing something.
Yes, representables are small. But I wouldn’t say it’s simple… there are a few details to be checked: that there is a category $Ind (\mathcal{C})$, that the objects in $Ind (\mathcal{C})$ are filtered colimits of representables, etc.
Thanks again. I checked that up (it is in SGA4). In my pro-category case, those facts (or their duals) are in the earlier parts of the monograph. For the Lab the DEFINITION of $Ind(\mathcal{C})$ puts these as prerquisites for the structure then derives the construction, which is fine and dandy but did not help me that much as the definition I had used was not in that form.
Looking at Chorny’s treatment of the Grossman-Isaksen model category structure on $Pro(\mathcal{S})$, he only uses that the representables / constant pro-objects are cosmall.
(I must try to work out how this all ties in with Algebraic WFSs à la Garner! I have Isaksen, Chorny, Barnea-Schlank and also the AWFS stuff from Richard and Emily. It should make a neater way to view the pro-homotopy theory stuff that seems to be important for motives etc.)
There are some nice things you can say about algebraically free AWFSes (that are generated by some small set) on locally presentable categories. Also, John Bourke and Richard Garner have a theorem essentially saying that (under certain hypotheses) an accessible AWFS is cofibrantly generated (in a generalised sense) by a small double category.
My problem would be the dual. Do you have a reference for that? As locally presentable categories are almost Ind-categories, so that looks interesting.
Indeed, a locally finitely presentable category is necessarily equivalent to $Ind (\mathcal{C})$ for some small $\mathcal{C}$ with finite colimits.
I worked out some properties of algebraically free AWFSes for an essay: see section 2.2 here. It’s mostly small improvements on the folklore. The theorem of Bourke and Garner I mentioned is not yet published – I heard of it when John gave a seminar talk on it – but perhaps you could write to one of them for details.
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