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][1])

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, and $Sp^X$ denotes the full subcategory of compact object in  the infinity category $Sh(X,Sp)$ of 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:** Given direct (1-)categories $I$ and $J$, 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 level Euler Characteristic of $F(X)$ from the levelwise Euler characteristic of $X$, when $X$ is a finitely presentable objects in $\widehat{I}$. This construction is quite usefull 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 what I want to construct was closely related to the $K$-theory or topological Hochshild homology of the category of compact presheaves of spectra on $I$, and I was hoping to use some well established and more general machinery to define the invariant I'm after. 

To some extent taking $THH$ actually is "good enough" for most of the applications I have in mind. But I would really prefer if I could have a version of this invariant that get ride of this loop space...


  [1]: https://arxiv.org/pdf/1001.2282.pdf