TITLE: Finite limits in the category of smooth manifolds?
QUESTION [25 upvotes]: 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?

REPLY [2 votes]: 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.)