Sorry the counterexample doesn't work as I missed a minus sign, it's T-1= 1-T which is not a contradiction.

Oh it's actually simpler we can just take T:=1+1+1+... Then T+1=T so if T has an additive inverse 1=0 which means X=0.

@SaalHardali The category of sets and relations is such a category.

@TylerLawson People write $\widetilde{SL}_n$ most often. For n=2 it's the metaplectic group. You probably won't get better names because these are not matrix groups. I haven't seen notation for $GL_n$.

@SallHardali $Pr^L$ is an example. Or categories with countable coproducts (and the obvious maps).

Spaces and spans between them is an example

There's an unfortunate terminological class with $n$-semiadditivity in the sense of Hopkins and Lurie, which has a $n=\infty$ case...

that should be "terminological clash"

Is there any relationship between profinite homotopy theory and $G$-equivariant homotopy theory where $G$ is a profinite group?

That should also be @SaalHardali rather than @SalHardali

@TimCampion @AlexanderCampbell These are cool examples. Regarding the presentable categories, does it satisfy this semi-additivity for every ordinal? Or is it something special about $\omega$?

@SaalHardali Nothing special about $\omega$.

+

@TimCampion What's a good reference for this?

@SaalHardali Well, you know that limits in $Pr^L$ and $Pr^R$ are both computed as in the underlying category. This tells you that the terminal category is a zero object in $Pr^L$. It also tells you that products and coproducts in $Pr^L$ are both computed as the product of underlying categories, and the canonical map you can construct is an equivalence.

I guess I'm not sure what else to say.

Well

A related example is the 2-category of categories with profunctors between them

For similar reasons

This was one property of bimodule 2-categories used by Street in his characterization of them.

I also seem to recall this being commented on in some paper on proarrow equipments -- the phrase "structural form of additivity", which now I can't seem to find

oh

that phrase appears in the Street paper I already linked to, "Cauchy Characterization of Enriched Categories"

Has anybody tried to replicate Kelly's Unified Treatment of Transfinite constructions paper infinity-categorically?

@TimCampion If memory serves me right, at last year’s Octoberfest, Mathieu Anel indicated that he had generalised some portion of it, specifically those parts relating to the orthogonal subcategory problem.

A video of his talk is available here: ct-octoberfest.github.io

@AlexanderCampbell Cool, thanks!

That paper is hard for me to read because it works at such a high level of generality -- just figuring out how to specialize results to some basic cases takes some work. I think I'd actually be happy to understand everything just in the case where everything is presentable and the factorization systems he uses are trivial.

In this case, most of the paper would be overkill, but I still want to understand the basic maneuver better -- reducing from from free monads to free well-pointed endofunctors.

Of course, in the simple case I have in mind I suppose the orthogonal subcategory problem is already understood in HTT

That basic maneuver should be quite formal and should generalize nicely to $\infty$-categories, I hope -- if I could just get my head around it!

@TimCampion Let me recommend the Appendix of this paper of John Bourke: arxiv.org/abs/1712.02523

It gives a neat construction of free algebras for a pointed endofunctor.

Thanks!