The category of smooth manifolds does not have all finite limits. However, it does have some limits: it has finite products, it has splittings of idempotents, and it has certain other limits if we require some or all of the maps involved to be submersions. What is the most general thing we can say about what limits it does have, and what is the best reference for such limits?
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asked Apr 6, 2014 at 4:09
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1$\begingroup$ We also have pullbacks for transverse maps, so- more generally- if you write the canonical presentation of a limit as a reflexive equalizer of some products, one could present the equalizer as a pullback and ask if the given maps are transverse. But I don't think this is the most general thing one can say... $\endgroup$– Dylan WilsonCommented Apr 6, 2014 at 6:06
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$\begingroup$ What does it mean, in terms of the maps f and g that you're taking the equalizer of, for the maps in the resulting pullback to be transverse (i.e. for (f,g) to be transverse to the diagonal)? $\endgroup$– Mike ShulmanCommented Apr 7, 2014 at 13:38
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1 Answer
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In Definition 2.1 and Assumptions 2.2 of my paper http://arxiv.org/abs/math/0603563, I axiomatized what I found to be the most important properties of the category of manifolds.
These are:
$\blacktriangleright$ The category is equipped with a Grothendieck pretopology, i.e., a collection of morphisms called ``covers'', subject to the following three axioms: (1) Isomorphisms are covers. (2) The composition of two covers is a cover. (3) The pullback along a cover always exists, and is a cover.
In the case of manifolds, the ``covers'' are the surjective submersions.
$\blacktriangleright$ The category has a terminal object, and any map to the terminal object is a cover.
$\blacktriangleright$ Idempotents split.
$\blacktriangleright$ The retract of a cover is a cover.
$\blacktriangleright$ The Grothendieck topology is subcanonical (the representable functors $T → \mathrm{Hom}(T, X)$ are sheaves.)
answered Dec 27, 2015 at 21:36
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$\begingroup$ This mentions the finite limits, split idempotents, and pullbacks of surjective submersions, but nothing else that I can see, e.g. not pullbacks of non-surjective submersions, or pullbacks of pairs of transverse maps neither of which is a submersion. $\endgroup$ Commented Dec 28, 2015 at 4:26
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$\begingroup$ You are right. There is no way that I can see of recovering the notion "pair of transverse maps" given the notion "(surjective) submersion". So my answer is not a very good answer... Sorry for that. $\endgroup$ Commented Dec 28, 2015 at 18:11