Enriched category: Difference between revisions

Latest revision as of 15:08, 29 December 2009

Suppose $M$ is a monoidal category. A category enriched in $M$ is defined as a $C$ equipped with the following data:

A collection of objects $O b C$ .
For any objects $A, B \in O b C$ , an object $C (A, B) \in O b M$ .
For any object $A \in O b C$ , a unit map $u_{A}$ from the unit object in $M$ to $C (A, A)$ .
For any objects $A, B, C \in O b C$ , a morphism in $M$ :

$\circ : C (B, C) \otimes C (A, B) \to C (A, C)$

where $\otimes$ denotes the monoidal operation on $M$ .

satisfying the following conditions:

For any $A, B, C, D \in O b C$ , with $f \in C (A, B), g \in C (B, C), h \in C (C, D)$ , we have $h \circ (g \circ f) = (h \circ g) \circ f$ .
For any $A, B \in O b C$ , the map $\circ : C (B, B) \otimes C (A, B) \to C (A, B)$ , composed with $u_{B}$ on the first coordinate, gives the identity map on the second coordinate. Similarly, the other way around. Need to make more precise -- Fill this in later

@@ Line 8: / Line 8: @@
 * For any objects <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a morphism in <math>\mathcal{M}</math>:
-<math>\circ: \mathcal{C}(B,C) \otimes \mathcal{C}(A,B) \to \mathcal{C}(A,B)</math>
+<math>\circ: \mathcal{C}(B,C) \otimes \mathcal{C}(A,B) \to \mathcal{C}(A,C)</math>
 where <math>\otimes</math> denotes the monoidal operation on <math>\mathcal{M}</math>.
@@ Line 15: / Line 15: @@
 * For any <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
-* For any <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, {{fillin}}
+* For any <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, the map <math>\circ: \mathcal{C}(B,B) \otimes \mathcal{C}(A,B) \to \mathcal{C}(A,B)</math>, composed with <math>u_B</math> on the first coordinate, gives the identity map on the second coordinate. Similarly, the other way around. Need to make more precise -- {{fillin}}