Maybe there is no good answer to this, but I'm trying to get a feel for what the $K$-theory of a (permutative or symmetric monoidal $\infty$-)category computes.


In algebraic $K$-theory, we have explicit descriptions for the low $K$-groups, where $K\_0(R)$ is the Grothendieck group of finite projective modules and $K\_1(R)$ is the abelianization of $\text{Gl}(R)$ (the infinite general linear group) so for commutative rings we compute the units of the ring.


The $K$-theory of $\mathrm{Proj}\_R$ recovers $K(R)$, and the description of $K\_0$ translates to the categorical generalization, that is $K\_0(\mathcal{C})$ is the Grothendieck group of $\mathcal{C}$. However I could not find an analogue of $K\_1(\mathcal{C})$, so to speak what are the units of a category? Maybe the automorphisms of the unit object?



$K_1$ of Categories for intuition
For an exact category ${\cal N}$, Dan Grayson (who occasionally shows up on MathOverflow) gave explicit generators and relations for $K\_n$ [here](https://www.ams.org/journals/jams/2012-25-04/S0894-0347-2012-00743-7/S0894-0347-2012-00743-7.pdf).


What follows is essentially quoted directly from that paper except that I've done some minor edits to make it self-contained:



> 
> **Definition.** An *acyclic binary multicomplex of dimension $n$* in ${\cal N}$ is a bounded ${\mathbb Z}^n$-graded object of ${\cal N}$ equipped with acyclic differentials $d\_i$ and $d\_i'$ in direction $i$, for each $i$ with $1 \le i \le n$, where the maps $d\_i$ and $d\_i'$ commute with $d\_j$ and $d\_j'$ provided $i \neq j$. (Four pairs of maps have just been asserted to commute: $d\_i$ commutes with $d\_j$ , $d\_i$ commutes with $d\_j'$ , $d\_i'$ commutes with $d\_j$, and $d\_i'$ commutes with $d\_j'$.) 
> A map between acyclic binary multicomplexes is a map of underlying graded objects that commutes with each of the differentials, and a short exact sequence is a short sequence consisting of short exact sequences in each component.
> 
> 
> **Theorem.** If ${\cal N}$ is an exact category, then the abelian group $K\_n({\cal N})$ has the following presentation by generators and relations:
> 
> 
> There is one generator for each acyclic binary multicomplex of dimension $n$ in ${\cal N}$ .
> 
> 
> The relations are of two types:
> 
> 
> (1) those coming from short exact sequences, as in the definition of the Grothendieck group;
> 
> 
> (2) any generator that is trivial, in the sense that $d\_i = d\_i'$ for some $i$, is made to vanish.
> 
> 
> 


In particular, the generators for $K\_1$ are *a priori* ${\mathbb Z}$-indexed sequences of objects equipped with two acyclic differentials. But from earlier work of Nenashev, we can restrict to a smaller group of generators, namely the *doubly exact sequences* in which all but three successive objects vanish. That is, a generator is a pair of short exact sequences having the same objects in all three non-zero locations. (This applies only to $K\_1$, not to higher $K\_n$.)


