# Pullbacks for primitive recursive functions.

Since a pullback of two functions f and g with common codomain into Set category is just a subset of cartesian product like this: {(x,y)/f(x)=g(y)} (with two more functions not important here) could this pullback set be the empty set in some cases (for exemple in the case of constant functions)?

My question is related to find pullbacks for primitive recursive functions, where the functions are all of them total and the domain and codomain are powers of natural numbers set. Which could be the aspect of those supposed pullbacks for constant functions since the empty set is not there available? Do they exist?

• "could this pullback set be the empty set in some cases?" Yes! Pullback to a point is just "fibre of the map" so sure a fibre can be empty. Feb 25, 2010 at 15:24
• Pullback to a point? You mean a degenerate case of a pullback? Makes it any sense? And..what if we haven't the empty set available? Feb 25, 2010 at 15:37
• Are you asking about the existence of pullbacks in the Lawvere Theory of primitive recursive functions? ncatlab.org/nlab/show/Lawvere+theory Feb 25, 2010 at 15:40
• Ximo, it looks to me as if your first paragraph is asking the following question. (If not, please clarify.) Let X, Y and Z be sets, and let f: X --> Z and g: Y --> Z be functions. Is it possible that the pullback of f with g is empty? Kevin answered "no, e.g. take X to be the one-element set". More specifically, you could take X and Y both to be one-element sets, Z to be a two-element set {z_1, z_2}, and f and g to be the functions with respective images z_1 and z_2. Feb 25, 2010 at 16:22
• Mh. In that case the square giving raise to the pullback woludn't be commutative and therefore the pullback set would be empty. Isn't it? My question is: could we make that for prim. rec. functions (where all of them are total and we've not the empty set as a possible domain)? Have they pullbacks? F.G.Dorais: I wish I could find that info in the web you send, but I don't see it there. Thank you anyway. Feb 25, 2010 at 16:34