Definition. Hom candidates [frct-001C]
Definition. Hom candidates [frct-001C]
For any \(x\in B\) and displayed objects \(u,v\in E_{x}\), we define a hom candidate for \(u,v\) to be a span \(u\leftarrow \bar {h} \rightarrow v\) in \(E\) in which the left-hand leg is cartesian:
In the above, \(h\) should be thought of as a candidate for the “hom object” of \(u,v\), and \(\epsilon _{h}\) should be viewed as the structure of an “evaluation map” for \(h\). This structure can be rephrased in terms of a displayed category \(\mathbf {H}_{E_{x}}(u,v)\) over \({B}_{/x}\):
- Given \(h\in {B}_{/x}\), an object of \(\mathbf {H}_{E_{x}}(u,v)_{h}\) is given by a hom candidate whose apex in the base is \(h\) itself. We will write \(\bar {h}\) metonymically for the entire hom candidate over \(h\).
Given \(\alpha :l\to h\in {B}_{/x}\) and hom candidates \(\bar {l}\in \mathbf {H}_{E_{x}}(u,v)_{l}\) and \(\bar {h}\in \mathbf {H}_{E_{x}}(u,v)_{h}\), a morphism \(\bar {h}\xrightarrow [\alpha ]{} \bar {l}\) is given by a cartesian morphism \(\bar \alpha :\bar {l}\xrightarrow [\alpha ]{}\bar {h}\) in \(E\) such that the following diagram commutes:
