Goto Chapter: Top 1 2 Ind

### 2 Category of presheaves (with bounds) of a (linear) category

Here we assume that the object set of the source category carries a discrete total order. This code is temporary and should at some point in the future be replaced by something like "presheaves with bounded support".

#### 2.1 GAP categories

##### 2.1-1 IsPreSheafWithBoundsCategory
 ‣ IsPreSheafWithBoundsCategory( category ) ( filter )

Returns: true or false

The GAP category of a presheaf category.

##### 2.1-2 IsCellInPreSheafWithBoundsCategory
 ‣ IsCellInPreSheafWithBoundsCategory( cell ) ( filter )

Returns: true or false

The GAP category of cells in a presheaf category.

##### 2.1-3 IsObjectInPreSheafWithBoundsCategory
 ‣ IsObjectInPreSheafWithBoundsCategory( obj ) ( filter )

Returns: true or false

The GAP category of objects in a presheaf category.

##### 2.1-4 IsMorphismInPreSheafWithBoundsCategory
 ‣ IsMorphismInPreSheafWithBoundsCategory( mor ) ( filter )

Returns: true or false

The GAP category of morphisms in a presheaf category.

#### 2.2 Attributes

##### 2.2-1 Source
 ‣ Source( PSh ) ( attribute )

Returns: a CAP category

The source category C of the presheaf category PSh=PSh(C,V).

##### 2.2-2 Target
 ‣ Target( PSh ) ( attribute )

Returns: a CAP category

The target category V of the presheaf category PSh=PSh(C,V).

##### 2.2-3 OppositeOfSource
 ‣ OppositeOfSource( Hom ) ( attribute )

Returns: a CAP category

The opposite C^\mathrm{op} of the source category C of the presheaf category PSh=PSh(C,V).

##### 2.2-4 Source
 ‣ Source( F ) ( attribute )

Returns: a CAP category

The source of the presheaf F.

##### 2.2-5 Target
 ‣ Target( F ) ( attribute )

Returns: a CAP category

The target of the presheaf F.

##### 2.2-6 YonedaEmbedding
 ‣ YonedaEmbedding( B ) ( attribute )

Returns: a CAP functor

##### 2.2-7 YonedaEmbeddingOfSourceCategory
 ‣ YonedaEmbeddingOfSourceCategory( PSh ) ( attribute )

Returns: a CAP functor

##### 2.2-8 CoYonedaLemmaOnObjects
 ‣ CoYonedaLemmaOnObjects( F ) ( attribute )

##### 2.2-9 CoYonedaLemmaOnMorphisms
 ‣ CoYonedaLemmaOnMorphisms( phi ) ( attribute )

#### 2.3 Constructors

##### 2.3-1 PreSheavesWithBounds
 ‣ PreSheavesWithBounds( B, C ) ( operation )
 ‣ PreSheavesWithBounds( B ) ( operation )

Returns: a CAP category

Construct the category Hom( B^op, C ) of functors from the opposite of the small category B to the category C as objects and their natural transformations as morphisms.

#### 2.4 Operations

##### 2.4-1 ApplyObjectInPreSheafCategoryToObject
 ‣ ApplyObjectInPreSheafCategoryToObject( F, obj ) ( operation )

Returns: a CAP object

Apply the presheaf F to the object obj. The shorthand is F(obj).

##### 2.4-2 ApplyObjectInPreSheafCategoryToMorphism
 ‣ ApplyObjectInPreSheafCategoryToMorphism( F, mor ) ( operation )

Returns: a CAP morphism

Apply the presheaf F to the morphism mor. The shorthand is F(mor).

##### 2.4-3 ApplyMorphismInPreSheafCategoryToObject
 ‣ ApplyMorphismInPreSheafCategoryToObject( eta, obj ) ( operation )

Returns: a CAP morphism

Apply the presheaf morphism eta to the object obj. The shorthand is eta(o).

Goto Chapter: Top 1 2 Ind

generated by GAPDoc2HTML