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Consider the following category of relations whose
objects are pairs $(X, \rho_X),$ where $\rho_X\subseteq X\times X$ and $X$ is a set
morphisms are relation-preserving functions, i.e., $X\xrightarrow{f}Y$ such that if $(a,b)\in\rho_X$ then $(f(a), f(b))\in \rho_Y.$
This category is different f...
In category theory, 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.
== Definition ==
Formally, we start with a category C with finite products (i.e. C has a terminal object 1 and any two objects of C have a product). 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...
