What's the name of an object in the slice category?

Question

What do you call an object of the slice category? I have been calling them "slices" but this seems to be wrong: it seems that people use the word "slice" as a synonym of "slice category", not as a name for their objects.

Edit: The most generally applicable term for an object of $\mathcal C / X$ seems to be "object over $X$". This has a few drawbacks:

• It requires mentioning $X$ all the time, which gets annoying when we are working in a slice category over a fixed $X$ for a longer time,
• "Over $X$" is an adverbial clause, which makes it sound like it's a property, when it's actually structure. An object over $X$ is not an object of $\mathcal C$. The phrase "a morphism of/between objects over $X$" (or even just "a morphism over $X$"?) does not really ring the bell that the forgetful functor $\Sigma_X : \mathcal C / X \to \mathcal C$ may not be full.

Answer 1

As the nlab article already indicates: objects of $\mathcal{C}/X$ are called objects over $X$, or more precisely $\mathcal{C}$-objects over $X$. They are typically written as $$\begin{array}{c} Y \\ \downarrow \\ X \end{array}$$ See also Stacks/001G. Similarly, objects of $X / \mathcal{C}$ are called objects under $X$.

Answer 2

When $\mathcal C$ is a category of spaces (in the broadest sense), then objects in the slice category $\mathcal C/X$ are often called $X$-bundles. This is especially so in algebraic geometry and in topology. The idea is that an object $p:Y\to X$ is a collection of fibers $Y(x)=p^{-1}(x)$ indexed by the base space $X$. You can draw $Y\to X$ as a bundle of lines (the fibers) each sitting over their corresponding $x\in X$. When you take this very seriously, then you end up with dependent type theory.

Edit: Here some more names which go into the same direction, but do not indicate that the space might be locally a projection map. You can call $X \to S$ also an $S$-indexed space, or an $S$-indexed family of spaces.

Answer 3

Many algebraic geometry papers take place in the slice category of schemes over some base scheme $S.$ In practice, "scheme over $S$" is used very often in these contexts, but "$S$-scheme" is a common abbreviation, as is $k$-scheme for a $\mathrm{Spec}(k)$-scheme when $k$ is a commutative ring, especially a field. In my experience, there is never any abbreviation that throws out the $S$ while emphasizing the slice, because that's not how these objects are thought of. This terminology could be exported to many contexts: if the objects of $\mathcal C$ are widgets, then an object of $\mathcal C/X$ could readily be called $X$-widgets, though the familiarity of the language will vary with context.