Todd Trimble
Karoubi envelope

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

Note in particular that the identity on (c,e)(c, e) is given by e:cce: c \to c (not 1 c1_c). The inclusion i:CC¯i: C \to \bar{C} that takes cc to (c,1 c)(c, 1_c) and f:ccf: c \to c' to f:(c,1 c)(c,1 c)f: (c, 1_c) \to (c', 1_{c'}) is full and faithful. Any idempotent e:cce: c \to c in CC has a formal splitting given by e:(c,1 c)(c,e)e: (c, 1_c) \to (c, e) followed by e:(c,e)(c,1 c)e: (c, e) \to (c, 1_c).

Proposition

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

Proof

Each functor F:CDF: C \to D has an extension F¯:C¯D\bar{F}: \bar{C} \to D that is unique up to isomorphism. First, any such functor F¯\bar{F} preserves splittings of idempotents and thus must take the formal splitting (c,1 c)e(c,e)e(c,1 c)(c, 1_c) \stackrel{e}{\to} (c, e) \stackrel{e}{\to} (c, 1_c) in C¯\bar{C} to some splitting F(c)rdjF(c)F(c) \stackrel{r}{\to} d \stackrel{j}{\to} F(c) of F(e)F(e) in DD, and any choice of retraction-inclusion data (d,r,j)(d, r, j) is unique up to isomorphism. Supposing given such choices (d,r,j),(d,r,j)(d, r, j), (d', r', j') for (c,e),(c,e)(c, e), (c', e'), the definition of F¯(f)\bar{F}(f) for f:(c,e)(c,e)f: (c, e) \to (c', e') is forced to be the unique g:ddg: 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:C¯DF, G: \bar{C} \to D, a natural transformation θ:FG\theta: F \to G is uniquely determined by its restriction θi:FiGi\theta i: F i \to G i: for each object (c,e)(c, e) there exists a unique θ(c,e):F(c,e)G(c,e)\theta (c, e): F(c, e) \to G(c, e) that fits into naturality squares

F(c,1 c) F(e) F(c,e) F(e) F(c,1 c) θi(c) θ(c,e) θi(c) G(c,1 c) G(e) G(c,e) G(e) G(c,1 c)\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