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In , a branch of mathematics, a subobject is, roughly speaking, an that sits inside another object in the same . The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory, and subspaces from topology. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism that describes how one object sits inside another, rather than relying on the use of elements. The concept to a subobject is a . This generalizes concepts such as quotient sets, quotient groups, quotient spaces, quotient graphs, etc. An appropriate categorical definition of "subobject" may vary with context, depending on the goal. One common definition is as follows. In detail, let be an object of some category. Given two monomorphisms with codomain , we define an equivalence relation by if there exists an isomorphism with . Equivalently, we write if factors through —that is, if there exists such that . The binary relation defined by is an equivalence relation on the monomorphisms with codomain , and the corresponding equivalence classes of these monomorphisms are the subobjects of . The relation ≤ induces a partial order on the collection of subobjects of . The collection of subobjects of an object may in fact be a proper class; this means that the discussion given is somewhat loose. If the subobject-collection of every object is a set, the category is called well-powered or, rarely, locally small (this clashes with a different usage of the term , namely that there is a set of morphisms between any two objects). To get the dual concept of quotient object, replace "monomorphism" by "epimorphism" above and reverse arrows. A quotient object of A is then an equivalence class of epimorphisms with domain A. However, in some contexts these definitions are inadequate as they do not concord with well-established notions of subobject or quotient object. In the category of topological spaces, monomorphisms are precisely the injective continuous functions; but not all injective continuous functions are subspace embeddings.

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Category of groups

In mathematics, the Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a . The study of this category is known as group theory. There are two forgetful functors from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from groups to . M has two adjoints: one right, I: Mon→Grp, and one left, K: Mon→Grp. I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid.

Subobject

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of , a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow f : X → Y such that for all objects Z and all morphisms g1, g2: Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below.