Do pullbacks exist in the category of sets and partial functions? 


Are the 'the same' as they are in Sets? That is, given two partial functions $f : A \to C$ and $g : B \to C$, is the pullback given by $\{ (a,b) \in A\times B ~|~ f(a)=g(b) \}$? 


If not, what is a simple description of the pullback?



Pullbacks in Category of Sets and Partial Functions
Pullbacks exists but are not what you describe.


The answer is as follows:


The category $\mathcal{C}$ of sets and partial functions is equivalent to the category of based sets and based functions, by sending the set $A$ to $A$ disjoint union a base point $\*$ and sending $f$ to the obvious based function which sends everything on which $f$ was not defined to the base-point.


The pullback in based sets are well-known and for example the product $\times$ in based sets translates back through this equivalence to $\mathcal{C}$ and becomes:


$A \times\_\mathcal{C} B \approx (A \times\_{\textrm{set}} B) \sqcup\_{\textrm{set}} A \sqcup\_{\textrm{set}} B$


From the purely sets and partial function point of view this is also explainable. Indeed, any morphism from $Z$ to this product is given by a choice for each point in $Z$ of either: a point in $A$ and a point in $B$, or a point in $A$, or a point $B$, or nothing.


