Lex objects in a 2-category

Lex objects in a 2-category

[Steven Vickers]

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An nLab page https://ncatlab.org/nlab/show/cartesian+object<https://url.au.m.mimecastprotect.com/s/JBM-C2xMRkUpN0RwEcnfqu5Tmha?domain=ncatlab.org> defines an object of X of a 2-category K to be “cartesian” iff the 1-cell X -> 1 and the diagonal X -> X^2 both have right adjoints. This assumes K has finite 2-products.

If, further, K has finite PIE-limits, hence finite powers (cotensors) X^C, then it seems reasonable to define X to be “lex” if every diagonal X -> X^C has a right adjoint. Unless I’ve made a mistake, in the 2-category of categories this characterises the lex categories.

Have these lex objects in a 2-category been studied?

Steve Vickers.




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Re: Lex objects in a 2-category

[Ross Street]

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Dear Steve

"Studied" is probably the wrong word for
http://science.mq.edu.au/~street/Sketch.pdf
which is
F. Complete objects relative to a theory (June 1976)
on my Publication page.
Others will have better suggestions.

Ross

On 30 Oct 2025, at 3:05 am, Steven Vickers <s.j.vickers.1@bham.ac.uk> wrote:

An nLab page https://ncatlab.org/nlab/show/cartesian+object<https://ncatlab.org/nlab/show/cartesian+object> defines an object of X of a 2-category K to be “cartesian” iff the 1-cell X -> 1 and the diagonal X -> X^2 both have right adjoints. This assumes K has finite 2-products.

If, further, K has finite PIE-limits, hence finite powers (cotensors) X^C, then it seems reasonable to define X to be “lex” if every diagonal X -> X^C has a right adjoint. Unless I’ve made a mistake, in the 2-category of categories this characterises the lex categories.

Have these lex objects in a 2-category been studied?

Steve Vickers.


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Re: Lex objects in a 2-category

[Joseph Helfer]

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Hi Steve,

I study these in quite a bit of detail in my paper Internal 1-topoi in 2-topoi<https://url.au.m.mimecastprotect.com/s/73a5CgZ05JfAL4EGNSNfpu4N-mM?domain=arxiv.org>, but I was certainly not the first; for instance, in my Definition 2.14, I cite Street's Cosmoi of internal categories<https://url.au.m.mimecastprotect.com/s/4fmsCjZ12Rfn7ZoRAIRhVum6Q1i?domain=doi.org>, §9.14, where they are discussed.

Sincerely,
Joj

On Wed, Oct 29, 2025 at 4:32 PM Steven Vickers <s.j.vickers.1@bham.ac.uk<mailto:s.j.vickers.1@bham.ac.uk>> wrote:
An nLab page https://ncatlab.org/nlab/show/cartesian+object<https://url.au.m.mimecastprotect.com/s/rlDhCk815RCOMo254tQiVuGcVod?domain=ncatlab.org> defines an object of X of a 2-category K to be “cartesian” iff the 1-cell X -> 1 and the diagonal X -> X^2 both have right adjoints. This assumes K has finite 2-products.

If, further, K has finite PIE-limits, hence finite powers (cotensors) X^C, then it seems reasonable to define X to be “lex” if every diagonal X -> X^C has a right adjoint. Unless I’ve made a mistake, in the 2-category of categories this characterises the lex categories.

Have these lex objects in a 2-category been studied?

Steve Vickers.




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Re: Lex objects in a 2-category

[Steven Vickers]

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Thanks, Ross.

That does look useful in what I'm trying to do: given the notion of lex object X, I was looking for the object of models Mod(T, X) for each lex theory T (= finite limit = cartesian (Elephant) = essentially algebraic = partial Horn = quasiequational). (The next step is to consider those X for which for every theory morphism F: T -> T', Mod(F, X) has a left adjoint. I sense that there is there a useful notion of arithmetic-universe-object.)

If you forget D and t, T looks very like a limit sketch. C is presented by the graph (sorts and operators) and commutative diagrams (equations), while the objects b of B are for the cones. v applied to the fibre of u over b gives the diagram, w(b) the vertex, and τ provides the cone itself. Hence all that is still needed is to require that the cones are limit cones. As far as I can see, that is what the diagram at the top of p.2 is saying in the case t=1.

I think, therefore, there should be enough there to give me the objects of models that I'm looking for. Is there a publication I can reference?

However, I'm still puzzled by the role of t. Can you give any intuitions about just what kind of theory the general T presents? Let's say, in the finite case and for X with equalizers and finite products. Is it any more general than lex theories?

All the best,

Steve.

________________________________
From: Ross Street <ross.street@mq.edu.au>
Sent: Wednesday, October 29, 2025 9:57 PM
To: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>
Cc: Categories mailing list <categories@mq.edu.au>
Subject: Re: Lex objects in a 2-category

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Dear Steve

"Studied" is probably the wrong word for
http://science.mq.edu.au/~street/Sketch.pdf<https://url.au.m.mimecastprotect.com/s/C-unC5QP8ySZVngjNIzflCkftgd?domain=urldefense.com>
which is
F. Complete objects relative to a theory (June 1976)
on my Publication page.
Others will have better suggestions.

Ross

On 30 Oct 2025, at 3:05 am, Steven Vickers <s.j.vickers.1@bham.ac.uk> wrote:

An nLab page https://ncatlab.org/nlab/show/cartesian+object<https://url.au.m.mimecastprotect.com/s/8M15C71R63CAOPEp7hBi6Cojyw5?domain=urldefense.com> defines an object of X of a 2-category K to be “cartesian” iff the 1-cell X -> 1 and the diagonal X -> X^2 both have right adjoints. This assumes K has finite 2-products.

If, further, K has finite PIE-limits, hence finite powers (cotensors) X^C, then it seems reasonable to define X to be “lex” if every diagonal X -> X^C has a right adjoint. Unless I’ve made a mistake, in the 2-category of categories this characterises the lex categories.

Have these lex objects in a 2-category been studied?

Steve Vickers.



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Re: Lex objects in a 2-category

[Ross Street]

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Dear Steve

On 1 Nov 2025, at 4:32 am, Steven Vickers <s.j.vickers.1@bham.ac.uk> wrote:

If you forget D and t, T looks very like a limit sketch. C is presented by the graph (sorts and operators) and commutative diagrams (equations), while the objects b of B are for the cones. v applied to the fibre of u over b gives the diagram, w(b) the vertex, and τ provides the cone itself. Hence all that is still needed is to require that the cones are limit cones. As far as I can see, that is what the diagram at the top of p.2 is saying in the case t=1.

Yes, that is correct. (I should have been less opaque!)
A limit sketch yields a T with no D or t (i.e. t = 1_B), and with B discrete.
I expect we obtain the same models by restricting u to be a fibration.

However, I'm still puzzled by the role of t. Can you give any intuitions about just what kind of theory the general T presents? Let's say, in the finite case and for X with equalizers and finite products. Is it any more general than lex theories?

I think I was trying to include the case where cones were replaced by the cylinders of Freyd-Kelly
[Categories of continuous functors I, JPAA (1972)] and the case of "rules" of John Isbell
[General functorial semantics. I. Amer. J. Math. 94 (1972), 535--596].
However, I cannot remember to what extent that works.

I think, therefore, there should be enough there to give me the objects of models that I'm looking for. Is there a publication I can reference?

Joseph Helfer's suggestions are relevant:
"in my [paper] Definition 2.14, I cite Street's Cosmoi of internal categories, §9.14, where they are discussed."
Perhaps also Jean Bénabou's [Théories relatives à un corpus. C. R. Acad. Sc. Paris, t. 281 (17 novembre 1975)].
Section 5 of my [Conspectus of variable categories, JPAA 21 (1981) 307--338] could be of some value
because categories A can be brought into a bicategory K by tensoring (copowering) A with the terminal object of K.

Ross



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