Norwegian Wikipedia has an article on: Although, technically, there is no difference between an individual diagram and a functor or between a scheme and a category , the change in terminology reflects a change in perspective, just as in the set theoretic case: An exact name can be given only if the mineralogical composition is known, which cannot be determined in the field.
Geologists worldwide accept the diagrams as a classification of igneous, especially plutonic rocks. QAPF diagrams are mostly used to classify plutonic rocks phaneritic rocks , but are also used to classify volcanic rocks if modal mineralogical compositions have been determined.
QAPF diagrams are not used to classify pyroclastic rocks or volcanic rocks if modal mineralogical composition is not determined, instead the TAS classification Total-Alkali-Silica is used. TAS is also used if volcanic rock contains volcanic glass such as obsidian. An exact name can be given only if the mineralogical composition is known, which cannot be determined in the field. The QAPF diagram has four minerals or mineral groups chosen as important cornerstones of the classification.
F and Q for chemical reasons cannot exist together in one plutonic rock. Other minerals may and almost certainly occur in these rocks as well but they have no significance in this classification scheme.
This diagram does not determine whether a rock is gabbro, diorite, or anorthosite. There are other criteria used to decide that. Note that this diagram is not used for all plutonic rocks. Ultramafic rocks are the most important plutonic rocks that have separate classification diagrams. One is most often interested in the case where the scheme J is a small or even finite category. A diagram is said to be small or finite whenever J is. A morphism of diagrams of type J in a category C is a natural transformation between functors.
One can then interpret the category of diagrams of type J in C as the functor category C J , and a diagram is then an object in this category.
A cone with vertex N of a diagram D: The constant diagram is the diagram which sends every object of J to an object N of C and every morphism to the identity morphism on N. The limit of a diagram D is a universal cone to D. That is, a cone through which all other cones uniquely factor.
If the limit exists in a category C for all diagrams of type J one obtains a functor. Dually, the colimit of diagram D is a universal cone from D. If the colimit exists for all diagrams of type J one has a functor. Diagrams and functor categories are often visualized by commutative diagrams , particularly if the index category is a finite poset category with few elements: The commutativity corresponds to the uniqueness of a map between two objects in a poset category. Conversely, every commutative diagram represents a diagram a functor from a poset index category in this way.
Not every diagram commutes, as not every index category is a poset category: Further, diagrams may be impossible to draw because infinite or simply messy because too many objects or morphisms ; however, schematic commutative diagrams for subcategories of the index category, or with ellipses, such as for a directed system are used to clarify such complex diagrams.