In , a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous. Formally, we start with a C with finite products (i.e. C has a terminal object 1 and any two of C have a ). A group object in C is an object G of C together with morphisms m : G × G → G (thought of as the "group multiplication") e : 1 → G (thought of as the "inclusion of the identity element") inv : G → G (thought of as the "inversion operation") such that the following properties (modeled on the group axioms – more precisely, on the definition of a group used in universal algebra) are satisfied m is associative, i.e. m (m × idG) = m (idG × m) as morphisms G × G × G → G, and where e.g. m × idG : G × G × G → G × G; here we identify G × (G × G) in a canonical manner with (G × G) × G. e is a two-sided unit of m, i.e. m (idG × e) = p1, where p1 : G × 1 → G is the canonical projection, and m (e × idG) = p2, where p2 : 1 × G → G is the canonical projection inv is a two-sided inverse for m, i.e. if d : G → G × G is the diagonal map, and eG : G → G is the composition of the unique morphism G → 1 (also called the counit) with e, then m (idG × inv) d = eG and m (inv × idG) d = eG. Note that this is stated in terms of maps – product and inverse must be maps in the category – and without any reference to underlying "elements" of the group object – categories in general do not have elements of their objects. Another way to state the above is to say G is a group object in a category C if for every object X in C, there is a group structure on the morphisms Hom(X, G) from X to G such that the association of X to Hom(X, G) is a (contravariant) functor from C to the . Each set G for which a group structure (G, m, u, −1) can be defined can be considered a group object in the category of sets.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.