A subobject of is an isomorphism class of monomorphisms into . A quotient object is dually an isomorphism class of epimorphisms out of .
An object in a category with a terminal object is well-pointed when every monomorphism through which all points factor is invertible. This definition concerns detection of proper subobjects by points, and does not designate a chosen basepoint. If there are no points, it asserts that every subobject is the whole object. In particular, the empty set is well-pointed.
In a complete category, the intersection of a set-indexed family of subobjects of is their wide pullback in a category over . The resulting morphism into is a monomorphism, and it factors through every member. In a well-powered category, even the intersection of all subobjects satisfying a specified property is small.
For a limit-preserving functor and an arrow , a supported subobject is through which factors after applying . In a complete well-powered domain, their intersection remains supported. At the resulting pair , every supported subobject of is invertible. This minimality makes injective on morphisms to each cogenerator.
A category is well-copowered if, for each object , the epimorphisms have only a set of equivalence classes, where two are equivalent through an isomorphism of their codomains compatible with the quotient maps. Equivalently, its opposite category is well-powered.
An image of is the least subobject through which factors. In a category with pullback in a category constructions this gives a strong epimorphism–monomorphism factorization : pull back any mono in a lifting square for to ; minimality of the image forces the resulting mono to be invertible, producing a diagonal. Conversely, the lifting property of the strong part proves minimality. No stability under pullback is asserted; that is an additional property in a regular category.
For a morphism in an abelian category, its coimage is canonically isomorphic to its image . This yields an epimorphism followed by a monomorphism, unique up to unique compatible isomorphism. Every epi-mono factorization has this same middle object.
Pull back the two arrows of an image factorization in an abelian category. Monomorphisms are preserved by pullback, and pullback stability of epimorphisms in an abelian category preserves its epic part. Pasting the squares gives the pulled-back composite. The resulting epi-mono factorization is therefore its image factorization by uniqueness.
A commuting square induces a unique map between their images by and . The cokernel in a category property supplies existence, and cancellation of the epimorphic supplies uniqueness. These equations prove identity and composition laws, giving a functor from the arrow category to the abelian category.
In a category with pullback in a category constructions and image factorizations, direct image is a left adjoint to inverse image on subobjects. Frobenius reciprocity is . It holds for all such subobjects exactly when strong epimorphisms are stable under pullback along monomorphisms. For sufficiency pull the strong part of the image factorization back along the mono into its intersection with ; for necessity take and strong, whose image is the whole codomain.
A category is well-powered when every object has only a set of subobjects. It is well-copowered when every object has only a set of quotient objects.
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A subobject is a term used in various fields such as mathematics, computer science, and programming, and its meaning can vary depending on the context. Here are a few interpretations: 1. **Mathematics**: In category theory, a subobject is a generalization of the concept of a subset. It refers to a monomorphism (injective morphism) from one object to another, essentially capturing the notion of a "part" of an object in a categorical framework.