locally cartesian category

A category $C$ is **locally cartesian** if each of its slice categories $C/x$ is a cartesian monoidal category, meaning that $C/x$ has all finite products. Another way to say this is that $C$ has all finite fibred products or equivalently that $C$ has all pullbacks.

A **finitely complete category** is precisely a locally cartesian category that has a terminal object.

Here is some discussion on terminology from cartesian monad:

I would call a category with all pullbacks ‘locally cartesian’. Shouldn't a cartesian category at least have a terminal object? Would a terminal object make any difference here? —Toby

I think you’re right about the standard terminology. Both Tom Leinster and Jurgen Koslowski (pdf) use the terminology above. I’m not sure how we want to resolve this here. I’ll think about it, and look at the standard references, and fix the terminology on this page if nobody else does first. -Patrick

How is this? -Patrick

*Mike*: The Elephant definitely uses “cartesian” to mean “all finite limits.” However, I’m not sure how universal that is; I think at least in the past, some people have used “cartesian” to refer only to finite *products*. I’m sure that a cartesian object in a 2-category has been defined to be one such that $A\to A\times A$ and $A\to 1$ have right adjoints. Also the notion of “cartesian bicategory” refers only to finite products (and extra stuff too). And cartesian monoidal category and cartesian closed category certainly only means finite products. On the other hand, of course as you say, in this context “cartesian monads” only refer to pullbacks, not terminal objects and hence not products. There is also the use of “cartesian square” to mean a pullback square, which generalizes to “cartesian morphism” in a fibration.

*Toby*: Yes, despite the historical justification that Johnstone gives in the Elephant, I'd stick with (what I understand to be) the usual terminology: ‘cartesian’ means finite products. Then ‘locally cartesian’ means each slice is cartesian, hence pullbacks. Now the term for all finite limits is ‘cartesian locally cartesian’ (and topologists, at least, do say things like ‘connected locally connected’), but that mouthful just tells us that the time has come to say ‘left exact’ or ‘finitely complete’ instead.

*Mike*: I assume you mean that ‘cartesian’ means finite *products*. “Finitely complete” is also an reasonable term meaning “having finite limits.”

*Toby*: Yes, of course, fixed.

Revised on July 9, 2010 04:52:35
by Toby Bartels
(173.60.119.197)