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Algebraic $K$-theory of a symmetric monoidal category $C$ is defined in two steps: first take geometric realization, then group complete: $K(C) = \Omega B |C|$.

In HAK II, Grayson (following Quillen) lifts the group completion step to the categorical level, defining a symmetric monoidal category $C^{-1} C$ such that $K(C) = |C^{-1} C|$. So $C^{-1} C$ is a "pre group completion" in the sense that it realizes to group completion. But what is the significance of $C^{-1}C$ qua symmetric monoidal category, before geometric realization?

It's not the group completion at the categorical level. Although every object $c \in C$ has an "inverse" $c^{-1}$, and although there is an obvious map $I \to c \otimes c^{-1}$ (where $I$ is the monoidal unit), this map is not an isomorphism. In fact, there need not be a map in the other direction $c\otimes c^{-1} \to I$. Only upon geometric realization does the map $I \to c \otimes c^{-1}$ become invertible. So there remains the

Question: What is the universal property of $C^{-1} C$ as a symmetric monoidal category? EDIT What is the relationship between the construction $C \mapsto C^{-1} C$ and group completion at the level of categories?

Note that the nonexistence of a morphism $c \otimes c^{-1} \to I$ rules out something else: the free symmetric monoidal category with duals for objects on $C$ would also realize to $K(C)$, but that category would have both unit and counit maps, whereas $C^{-1}C$ has only unit maps.

It's probably worth recalling the construction. An object of $C^{-1} C$ (by which I really mean $Iso(C)^{-1} C$ in Grayson/Quillen's notation) is a pair of objects of $C$, which we may write $c_1^{-1} \otimes c_2$. A morphism $c_1^{-1} \otimes c_2 \to d_1^{-1} \otimes d_2$ is an equivalence class of triples $(c,f_1,f_2)$ where $c \in C$, $f_1: c_1 \otimes c \to d_1$ is an isomorphism in $C$, and $f_2: c_2 \otimes c \to d_2$ is a morphism in $C$; these are modded out by the diagonal action of $Aut_C(c)$. Defining composition can be done by following your nose.

Grayson/Quillen's construction actually applies more generally to any symmetric monoidal category $S$ acting on a category $C$, producing a category $S^{-1}C$, but I'd be happy with understanding the case $S = Iso(C)$, and perhaps even with the restriction that $C$ be a groupoid.

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    $\begingroup$ Thomason's paper "Beware the phony multiplication on Quillen's $\mathcal{A}^{-1}\mathcal{A}$" explains some things that look like they work, but are subtly wrong. $\endgroup$ Commented Oct 20, 2017 at 17:51
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    $\begingroup$ If I understand correctly, the universal property is that of the quotient of $C\times C$ by the diagonal action of $S=Iso(C)$; generally, given an action of a monoidal category $S$ on a category $C$, Grayson defines the quotient as a universal pair $(q,\alpha)$ where $q:C\to Q$ is a functor and $\alpha$ is an isomorphism between $q\circ\text{(action)}:S\times C\to C\to Q$ and $q\circ\text{(projection)}:S\times C\to C\to Q$. $\endgroup$ Commented Oct 20, 2017 at 19:33
  • $\begingroup$ @მამუკაჯიბლაძე Right... I suppose in that sense it's pretty straightforward. And this generalizes the usual set-level construction of the group completion of a commutative monoid. I suppose what's puzzling me is the fact that this construction coincides with group completion for sets and for spaces, but not for categories. Maybe I should formulate my question as follows: What is the relationship between this construction and group completion at the level of categories? $\endgroup$
    – Tim Campion
    Commented Oct 20, 2017 at 20:34

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It is the classifying category for the left action of $C$ on its product $C \times C$.

Let me elaborate on this a bit further. Let $C$ be an $E_\infty$-monoid, for example represented by a symmetric monoidal groupoid (which is the case for Quillen's applications of e.g. $Proj(R)$). We can then consider the symmetric monoidal $\infty$-category $B C$ given by a single object and mapping space equal to $C$.There's a natural functor $C^\circlearrowleft : B C \rightarrow Spc$, essentially given by the functor corepresented by the single object. Here $Spc$ means the $\infty$-category of spaces.

Then we can consider the functor $C^\circlearrowleft \times C^\circlearrowleft : B C \rightarrow Spc$, and take its unstraightening, i.e. Grothendieck construction. This is given by the functor $C^{-1} C \rightarrow B C$ which sends the arrow $s+ : (n,m) \rightarrow (s+n,s+m)$ in $C^{-1} C$ to the arrow $* \xrightarrow{s} *$ in $B C$.

An almost trivial consequence of this is that the realization of $C^{-1} C$ is just the ($\infty$-categorical) colimit $(C^\circlearrowleft \times C^\circlearrowleft)_{hC}$.

This construction exists for an arbitrary $E_n$-monoid, but needs Quillen's additional assumptions that we started with a symmetric monoidal groupoid on which addition by an object is faithful to make sure that $C^{-1} C$ is actually (equivalent to) a 1-category.

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