I'm wondering if there is an invariant, similar to algebraic K-theory, topological hochshild homologic, topological cyclic homology etc... that has the following properties:
a) It attach to each small stable $\infty$-category $A$ a spectrum $E(A)$.
b) Its functorial on exact functor.
c) It satisfies an additivity property similar to that of $K$-theory, $THH$, i.e. at the minimum it sends split exact sequence of small stable $\infty$-category to (split) cofiber sequences. (so essentially it is an "additive invariant" in the sense of Blumberg, Gepner, Tabuada)
d) For each object $a \in A$, there is an element $\chi(a) \in E(A)$ that is natural and behave additively on cofiber sequence, (so more formally, there is a natural transformation $K(A) \to E(A)$
e) If $X$ is a space (an $\infty$-groupoid), and $Sp^X$ denotes the full subcategory of compact object in the infinity category $Sh(X,Sp)$ of (locally constant) sheaves of spectrum on $X$, then:
$$E(Sp^X) \simeq \Sigma^\infty_+ X$$ is the suspension spectrum of $X$. Where the identification is functorial in $X$. I would also be happy if $E(Sp^X)$ gives the homology of $X$ instead, but the former would be more general.
Both $K$-theory, $THH$ and $TC$ have all these properties excepte maybe $(e)$. $THH$ fall close with $THH(Sp^X) \simeq \Sigma^\infty_+ \left( X^{S^1} \right)$.
I don't know well enough $TC$ and its variant to know directly if one them has property $(e)$. If it the case I would be very happy with a reference that proves it. Otherwise I'm hoping that some kind of "simplified K-theory" will do the trick, maybe a quotient of $K$-theory, but I don't know these topics well enough to figure it out.
Motivation: Let $I$ and $J$ be two finite direct (1-)categories. I denote by $\widehat{I}$ and $\widehat{J}$ the $\infty$-categories of presheaves of $\infty$-groupoids on them.
Given $F: \widehat{J} \to \widehat{J}$ a left adjoint functor preserving finitely presented objects, we can define a linear map: $$ |F| : \bigoplus_{i \in Ob(I)} \mathbb{Z} \to \bigoplus_{j \in Ob(J)} \mathbb{Z}$$
that "computes" the levelwise Euler characteristic of $F(X)$ from the levelwise Euler characteristic of $X$, when $X$ is a finitely presentable objects in $\widehat{I}$, in the sense that:
$$ \chi(X(i)) \in \bigoplus_{i \in Ob(I)} \mathbb{Z} \mapsto \chi(F(X)(j)) \in \bigoplus_{j \in Ob(j)}$$
This construction is quite useful when studying for example nice monads on such presheaf categories: it attach easy to compute and quite subtle numerical invariant to certain nice finitary functors.
I would like to upgrade this, by replacing abelian group by connective spectra and allowing $I$ and $J$ to be more general $\infty$-categories, but I'm struggling in proving that what I want to construct has the appropriate functorialy properties.
I realized at some point that functor above could be interpreted as:
$$ \pi_0 THH( Sp^{I^{op}}) = \bigoplus_{i \in I} \mathbb{Z} $$
where $Sp^{I^{op}}$ denotes the category of compact presheaves of spectrum on $I$. Indeed using additivity of $THH$, we can show by induction on height that:
$$ THH( Sp^{I^{op}}) = \Sigma_\infty^+ \left( Ob(I) \right)$$
To some extent taking $THH$ actually is "good enough" for most of the applications I have in mind. But it does not behave quite as I would like, and this makes everything more complicated. Typically, when $I$ is a generalized direct $\infty$-category (that is it can have some automorphisms) satisfying some finiteness condition I won't go into, I would like the invariant to be:
$$ \Sigma^\infty_+ ( core(I)) $$
where $core$ denotes the maximal sub $\infty$-groupoid. While, induction on height gives:
$$THH(Sp^{I^{op}}) = \Sigma^\infty_+ ( core(I)^{S^1}) $$
This of course is exactly because (e) fails for THH. But on the other hand, when I consider the example I'm interested in, it seems consistant that one could get rid of this $S^1$ : The way it appears in THH is because THH is the target of the universal trace map, and an endomorphism in $Sp^{I^{op}}$ can be over automorphisms of $I$ and hence remember a "loop" in $I$, but I' only interested in "characteristic of objects" (see point (d)) not trace of maps, so this "circle" is not relevant for me. So I'm hoping there is a different invariant from THH that would get rid of it.
I started studying the alternative (K-theory, TC etc...) but couldn't really decide if one of them was solving my problem... K-theory do not have this circle showing up, but lots of additional things, that are not relevant to me, (basicaly the K-theory of the point) comes up. And for TC (and its negative or periodic variant), I haven't been be able to understand how it behaves on spaces. I'm hoping an expert on the topic can point me to the right direction...
