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I'm looking for the name of a specific kind of "weak" pullback. I'm curious if anyone might recognize it. Consider a category $\mathbf{ParFun}$ whose objects are sets, and whose morphisms $f:X \to Y$ are partial functions. The composite $g \circ f$ is defined to be the partial function $g \circ f$ with domain $\{x \in \text{Dom}(f) \: : \: f(x) \in \text{Dom}(g) \}$.

To take the pullback of a cospan $f:X \rightarrow Z \leftarrow Y:g$, where $f,g$ have domains $D_f$ and $D_g$ respectively, the natural choice for the pullback is to take $P = \{(x,y) \in X \times Y \: : \: f(x) = g(y) \}$, and globally defined projection maps $\pi_X : P \to X$ and $\pi_Y : P \to Y$.

This is not a pullback square. To see this, take a new set A, and partial functions $\alpha: A \to X$ and $\beta:A \to Y$, with domains $D_\alpha$ and $D_\beta$, such that $f \pi_X = g \pi_Y$. One would like to define a map $\phi:D \to P$ by $\phi(x) = (\alpha(x), \beta(x))$. But what should the domain of $\phi$ be? The maximal domain for $\phi$ is $D_\alpha \cap D_{\beta}$, but it could be the case that this domain is strictly contained in at least one of $D_\alpha$ or $D_\beta$. If $D_\alpha \neq D_\beta$, then at least one of $\pi_X \phi \neq \alpha$ or $\pi_Y \phi \neq \beta$, meaning we have failed to find a pullback.

Nonetheless, this is clearly the "best choice" for a pullback square. Indeed, there is a unique "domain restriction" for $\alpha$ and $\beta$ that would make $\phi$ and isomorphism, and this restriction is "minimal" in some sense.

I think this could be related to the bicategory of partial maps, but I don't see a way to recognize this structure as a 2-pullback nor a comma-object. Nor do I exactly see how to formalize this as a property in a 2-category; the choice of $\phi$ is not unique in having the property that there are domain-restrictions for $\alpha$ and $\beta$, but it does lead to the largest domain for the restrictions of $\alpha$ and $\beta$. Is there perhaps some kind of 2-category pullback that I don't know about?

Joe
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1 Answers1

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I believe you are describing a restriction pullback in the sense of section 4.4 of Cockett and Lack's Restriction categories III: colimits, partial limits, and extensivity. In particular, they note that every restriction limit in a split restriction category becomes an actual limit in the subcategory of total maps, which in the case of the restriction category $\mathbf{Par}$ of sets and partial functors is the category of sets and total functions; indeed, by Proposition 4.12 ibid., $\mathbf{Par}$ has all restriction limits. (Note that $\mathbf{Par}$ is indeed split by Proposition 3.1 of Cockett and Lack's Restriction categories I: categories of partial maps, since it is a category of partial maps.)

varkor
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