F est une relation d'équilibre isomorphism. f is an isomorphism.
H σχέση της ισορροπίας
Η ιφιγένεια γεωργιάδου πρόσθεσε 125 νέες φωτογραφίες στο άλμπουμ: ΕΠΙΣΤΗΜΗ. 8 Μαρτίου 2014Universal constructions, limits, and colimits
Main articles: Universal property and Limit (category theory)
Using the language of category theory, many areas of mathematical study can be categorized. Categories include sets, groups and topologies.
Each category is distinguished by properties that all its objects have in common, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i.e., we do not know whether an object A is a set, a topology, or any other abstract concept. Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find universal properties that uniquely determine the objects of interest.
Indeed, it turns out that numerous important constructions can be described in a purely categorical way. The central concept which is needed for this purpose is called categorical limit, and can be dualized to yield the notion of a colimit. 27 Ιουλίου στις 12:44 μ.μ.