Todd Trimble Karoubi envelope

Let $C$ be a category. Recall that the Cauchy completion of $C$, denoted $\bar{C}$, may be described formally as follows:

• Objects are pairs $(c, e)$ where $e: c \to c$ is an idempotent of $C$.

• Morphisms $f: (c, e) \to (c', e')$ are morphisms $f: c \to c'$ such that $e' f = f = f e$; these are composed as in $C$.

Note in particular that the identity on $(c, e)$ is given by $e: c \to c$ (not $1_c$). The inclusion $i: C \to \bar{C}$ that takes $c$ to $(c, 1_c)$ and $f: c \to c'$ to $f: (c, 1_c) \to (c', 1_{c'})$ is full and faithful. Any idempotent $e: c \to c$ in $C$ has a formal splitting given by $e: (c, 1_c) \to (c, e)$ followed by $e: (c, e) \to (c, 1_c)$.

Proposition

Let $D$ be Cauchy-complete (as a $Set$-category). Then for any $C$, the restriction functor $[i, D]: [\bar{C}, D] \to [C, D]$ between functor categories is an equivalence.

Proof

Each functor $F: C \to D$ has an extension $\bar{F}: \bar{C} \to D$ that is unique up to isomorphism. First, any such functor $\bar{F}$ preserves splittings of idempotents and thus must take the formal splitting $(c, 1_c) \stackrel{e}{\to} (c, e) \stackrel{e}{\to} (c, 1_c)$ in $\bar{C}$ to some splitting $F(c) \stackrel{r}{\to} d \stackrel{j}{\to} F(c)$ of $F(e)$ in $D$, and any choice of retraction-inclusion data $(d, r, j)$ is unique up to isomorphism. Supposing given such choices $(d, r, j), (d', r', j')$ for $(c, e), (c', e')$, the definition of $\bar{F}(f)$ for $f: (c, e) \to (c', e')$ is forced to be the unique $g: d \to d'$ that is compatible with the retraction-inclusion data.

The preceding paragraph shows that that the restriction functor is surjective on objects (spurious reliance on AC can be fixed by using anafunctors). Moreover, for functors $F, G: \bar{C} \to D$, a natural transformation $\theta: F \to G$ is uniquely determined by its restriction $\theta i: F i \to G i$: for each object $(c, e)$ there exists a unique $\theta (c, e): F(c, e) \to G(c, e)$ that fits into naturality squares

$\array{ F(c, 1_c) & \stackrel{F(e)}{\to} & F(c, e) & \stackrel{F(e)}{\to} & F(c, 1_c) \\ \mathllap{\theta i(c)} \downarrow & & \downarrow \mathrlap{\theta(c, e)} & & \downarrow \mathrlap{\theta i(c)} \\ G(c, 1_c) & \stackrel{G(e)}{\to} & G(c, e) & \stackrel{G(e)}{\to} & G(c, 1_c) }$

so that the restriction functor is also full and faithful.

Created on August 8, 2014 at 04:33:23 by Todd Trimble