What is a quick way to show that every infty-category is the infty-categorical colimit over its category of simplices? Can one use the Reedy model structure on the category of simplices and show that the ordinary colimit computes the homotopy colimit?

@F.Abellan You can use the theory in Denis-Charles Cisinski's book "Les Préfaisceaux etc...". He shows that Δ is 'regular skeletal', which implies that every Cisinski-type model structure on Psh(Δ) has this property (what Cisinski calls 'regularity' i.e. every object is a hocolim of its category of elements).

This 'regular skeletality' is precisely related to this Reedy-style argument you're talking about (though it's done in Cisinski's book in much greater generality).

@HarryGindi sweet I'll have a look!

The chapter on catégories squelettiques (I think chapter 8) is the key input here.

Does Sp(C) admits a “natural” t-structure when C is presentable? Sp^cn(C) defines the connective part of a t-structure when C is an infinity-topos, but in general I’m skeptical. If it’s false, what is a counterexample?

I guess the notation Sp^cn(C) in the sense of homotopy groups is available only when C is an infinity-topos, so maybe I should write Sp^cn \otimes C

@NarukiMasuda Well, if C is already stable, then these are all giving you the same category.

I guess that's a natural t-structure, even if it isn't an interesting one.

@WilliamBalderrama I think this still holds when C is prestable, so my real question is a counterexample for additive+presentable->prestable.

@NarukiMasuda Abelian groups?

I suspect that the t-structure always exists (since you just take Sp^{cn}(C) as the colimit closure of the image of C→Sp(C), and that should always give you a t-structure since it's a localizing subcategory), it's just not always interesting...

@DenisNardin Ah, I see, I was expecting something like its homotopy category between presentable additive 1-category and presentable abelian 1-category but truncation is an obvious obstruction for the suspension to be fully faithful...

How is that a t-structure? I thought the point is that localizing subcategory is closed under extensions

closed under colimit+loop implies closed under extension by turning the triangle but I feel that's exactly the problem

@NarukiMasuda whoops sorry, I meant take the closure under colimits and extensions (and why not? It's always allowed...)

Of course in general it won't produce anything interesting, but it is a canonical t-structure (in fact the smallest t-structure such that the image of C lies in it)

I see, the connective part need not agree with Sp^cn\otimes C but it exists (the answer to the question I had in mind at first was negative anyway, thanks)

Well, along the original direction of counterexample I had in mind, I'm still interested in an example of a presentable 1-category whose abelian group objects do not form an abelian category.

(not a homotopy theory question, though)

what is the free monoidal category on a right-dualizable object? (it is not the tangle category, that corepresents "a fully dualizable object whose duals are fully dualizable, ad infinitum.)

@Dy

@DylanWilson @S.carmeli Thanks both, that reference clarifies quite a bit!