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*To*: categories@mta.ca*Subject*: categories: re: Pullback preserving Set-functors*From*: Peter Freyd <pjf@saul.cis.upenn.edu>*Date*: Mon, 11 Sep 2000 17:48:51 -0400 (EDT)*Sender*: cat-dist@mta.ca

Tobias Schroeder asks: Conjecture 1: Any Set-endofunctor that preserves kernel pairs [He called them "kernels"] preserves pullbacks. Counterexample: the functor that sends the empty set to the empty set and everything else to a fixed one-element set. (Note that this is also a counterexample for the conjecture-5 modification.) Conjecture 2: Any Set-endofunctor that preserves n kernel pairs *and inverse images* (i.e. pullbacks where one of the mapping is injective) preserves pullbacks. Proof: Given such a functor, T:A --> B, where A and B are sufficiently nice categories, lift it to T:A --> B/T1 to obtain a functor that not only preserves kernel-pairs and inverse images but the terminator. It suffices to prove that this lifted T preserves pullbacks (using the fact that the forgetful functor B/T1 --> B preserves pullbacks). hence it suffices to consider the conjecture assuming that the functor preserves also the terminator. A well-known argument then reduces the question to the preservation of binary products. (One may construct equalizers using inverse images and products and from there to arbitrary pullbacks is ancient. See 1.43 in Cats and Allegators.) What we will use is that the functor preserves inverse images and iterated products. Given sets X and Y we know that the product diagram: (X+Y)x(X+Y) / \ [add your own downwards arrowheads] X+Y X+Y is carried to a product diagram T((X+Y)x(X+Y)) / \ T(X+Y) T(X+Y). Using three inverse-image diagrams obtain: XxY / \ Xx(X+Y) (X+Y)xY / \ / \ X (X+Y)x(X+Y) Y \ / \ / X+Y X+Y Now apply T: T(XxY) / \ T(Xx(X+Y)) T((X+Y)xY) / \ / \ TX T((X+Y)x(X+Y)) TY \ / \ / T(X+Y) T(X+Y) As already observed, the very bottom / \ is a product diagram. Each rombus is an inverse-image pullback, hence preserved. Thus the top / \ / \ a product diagram. Conjecture 3: Same as Conj. 2 with *and inverse images* replaced by *and equalizers*. Inverse images of regular subobjects (that is, those that appear as equalizers) are regular: the equalizer of x,y under a map f is the equalizer of xf,yf. Hence if every subobject is regular than preservation of equalizers implies preservation of inverse images.

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