the suspension spectrum functor $\Sigma^\infty_+ : \mathrm{Spaces} \to \mathrm{Spectra}$ has a universal property: it's the initial exact functor to a stable $\infty$-category. more generally, as soon as $C$ has finite products, we get $\Sigma^\infty_+ : C \to \mathrm{Stab}(C)$. what further assumptions on $C$ do i need for this to have that same universal property?

Does 'exact' mean 'right exact'?

And do we at least know 'presentable' is enough?

How do you define Sigma^\infty_+ with just finite products?

@Dedalus if you just want "semiadditive" rather than "abelian", then a classic example is the category of suplattices. Similarly, another example would be the category of locally presentable categories (which is not locally small). "Abelianizing" one of these might give examples...

what's a moral reason why loop spaces of spheres appear when studying chromatic homotopy?

@TimCampion Thank you. In some sense, these seem a bit artificial, but maybe all examples will be of that form

@Dedalus Just as having biproducts (finite products and finite coproducts that coincide) implies (and is implied by, up to Cauchy completion) being enriched in commutative monoids, so too when K-ary products coincide with K-ary coproducts (K a cardinal), you have an enrichment in "K-ary commutative monoids," i.e. commutative monoids where infinite sums up to cardinality K are defined.

If you allow K to be the size of the universe, then I think this can only happen for a locally small category if the enrichment is in suplattices -- this is like the fact that a complete, small category is always a preorder.

If you want these commutative monoids to be abelian groups, then actually there may be an Eilenberg swindle argument that will tell you your homsets are trivial or something...

@DylanWilson yeah sorry, i meant "right-exact"

i do think "C is presentable" should suffice

The universal stable ∞-category with a right exact functor from Spaces is not Spectra, it's the Spanier-Whitehead ∞-category.

Spectra is the universal stable ∞-category with a left exact functor to Spaces.

@MarcHoyois by "spanier whitehead category", do you just mean suspension spectra and their desuspensions? in any case, i am looking for a universal property of $\Sigma^\infty_+ : \mathrm{Spaces} \to \mathrm{Spectra}$

i'm fine with saying "cocontinuous" or whatever

The Spanier-Whitehead ∞-category is not a subcategory of Spectra, but it maps to it, see SAG C.1.1.