Connected limits #
A connected limit is a limit whose shape is a connected category.
We give examples of connected categories, and prove that the functor given
(X × -) preserves any connected limit. That is, any limit of shape
J is a connected category is preserved by the functor
(X × -).
(Impl). The obvious natural transformation from (X × K -) to K.
(Impl). The obvious natural transformation from (X × K -) to X
Given a cone for (X × K -), produce a cone for K using the natural transformation
(X × -) preserves any connected limit.
Note that this functor does not preserve the two most obvious disconnected limits - that is,
(X × -) does not preserve products or terminal object, eg
(X ⨯ A) ⨯ (X ⨯ B) is not isomorphic to
X ⨯ (A ⨯ B) and
X ⨯ 1 is not isomorphic to