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?

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.

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.

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 group completion 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?

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, and $f_2: c_2 \otimes c \to d_2$ is a map; these are modded out by the diagonal action of $Aut_C(c)$. Defining composition can be done by following your nose.

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?

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.