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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeMar 27th 2013

The article synthetic differential ∞-groupoid claims that the site CartSp_synth of smooth loci of the form R^n×ℓW can also be constructed as the semidirect product CartSp⋉InfPoint. However, the cited article by Kock and Reyes has a corrigendum (http://numdam.org/item?id=CTGDC_1987__28_2_99_0) that claims that their original construction of the site CartSp_synth as a semidirect product is erroneous (the resulting category has the right objects, but too few maps). In which sense then the formula in the cited article should be interpreted? Or is it simply a reference to the original incorrect construction?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 27th 2013
• (edited Mar 27th 2013)

I think the definitions on all pages and files is and has always been stated correctly: the site is the full subcategory of that of smooth loci on those of the form CartesinanSpace-times-InfinitesimallyThickenedPoint.

I see that on the page you point to it says “semidirect product site” in quotation marks in the lead-in paragraph. Is that misleading? I always thought they should have just corrected their definition of what they mean by “semidirect product”, since it is actually a suggestive term. In the corrigendum article they suggest that one might say “twisted semirect product” instead, only then to decide not to say this at all.

It doesn’t really matter. We could just remove any mentioning of the word “semidirect product site”. I think it never appears in any formal definition, but just in text leading up to it. (If not, please alert me and I’ll change it.)

[edit: I made it read “twisted semidirect product” at synthetic differential infinity-groupoid now. But maybe I should just remove it alltogether. ]

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeMar 28th 2013

The semidirect product is also mentioned (as a formula) in the very first formula of Definition 2, which in fact reads CartSp_synthdiff := CartSp⋉InfPoint, and the := sign commonly means “is defined to be”.

I was just wondering whether and how one can actually define such semidirect products, especially since the naive semidirect product doesn't work, as pointed out by the corrigendum.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMar 28th 2013
• (edited Mar 28th 2013)

Okay, I have removed that symbol, not to be misleading.

I think of “(twisted) semidirect product” here as being just this: the full subcategory of some category on objects which are products of objects of two other (full) subcategories. The term serves no technical purpose except as to tell the reader: look, this is the site of definition of what elsewhere had already been called the “Cahiers topos”.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeMar 29th 2013

To me also, the phrase “semidirect product” suggests something constructed purely from the two input things, perhaps with an action of one on the other, rather than something of only part of which (such as the objects of a category) is constructed from those things.

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