Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?

2012-09-29T09:10:35Z

Motivation

In Pursuing Stacks, Grothendieck defines what he calls a basic localizer, which is, to put it roughly, a class of functors between small categories with which one can make homotopy in $Cat$. One axiom of basic localizers asserts that every category which has a terminal object is "aspherical", i.e. the canonical arrow from it to the point is in any basic localizer. It is a sufficient condition to get all the properties we want, provided the other axioms hold.

I have recently worked on "2-basic localizers", the analogous classes of (let say strict for convenience) 2-functors. It appears that what seems, from this viewpoint, the right analogous notion of "category with terminal object" is "2-category which has an object $z$ such that, given any object $x$, the category $Hom (x,z)$ has a terminal object". Instances of such 2-categories are "slices over an object". (There are, of course, three dual notions, the four of them corresponding to the two ways to slice over an object and the two ways to slice under an object.)

Question 1

Does this property have a standard name?

I am aware that there should be something like an adjunction between such a 2-category and the terminal one going on here, but I am really looking for standard terminology if there is one now.

Question 2

Do this kind of 2-categories or this kind of property appear naturally in other contexts?

This second question may be as silly as asking "where do categories with a terminal object crop up", but category theorists I have talked to do not seem to have encountered such a notion. I hope it could ring a bell for other people, especially those working on homotopy-related stuff.

EDIT: To be a bit more precise, I have the feeling that this property has something to do with prefibrations in $2-Cat$. Perhaps I will tell more about that later.

Answer by Michael Barr for Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?

2012-09-30T01:26:41Z

As a matter of general principle, homsets inherit properties of the target. This is not invariable, but just a guideline. It seems clear that in Cat, for every $z$ that has a terminal object, the category Hom$(x,z)$ will also have a terminal object since the functor that is constant at the terminal object will be terminal. I don't know if this is general (it is too late in the day) but that is where I would look.

Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object? - MathOverflow

Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?

Answer by Michael Barr for Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?