A category consists of objects, morphisms between them, associative composition, and an identity morphism on every object.
A functor maps objects and morphisms between categories while preserving identities and composition.
A covariant functor is representable when it is naturally isomorphic to for some object ; a contravariant functor is representable when it is naturally isomorphic to .
An epimorphism is right-cancellative: implies . In the Category of sets and in abelian groups, epimorphisms are precisely the surjective homomorphisms.
A subobject of is an isomorphism class of monomorphisms into . A quotient object is dually an isomorphism class of epimorphisms out of .
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.
An object is projective when every morphism lifts through every epimorphism . Equivalently, the functor preserves epimorphisms.
A finite limit is a categorical limit whose indexing category has finitely many objects and morphisms.
Limits of shape commute with colimits of shape when, for every , the canonical comparisonis an isomorphism whenever both sides exist.
A category is filtered when every finite diagram in it admits a cocone. Equivalently, it is nonempty, every pair of objects maps to a common object, and every pair of parallel arrows becomes equal after postcomposition.
A category is weakly filtered when every finite connected diagram in it admits a cocone. Equivalently, each of its connected components is a filtered category.
In the Category of sets, filtered colimits commute with finite limits. Elements in a finite limiting diagram involve only finitely many representatives and finitely many equalities, all of which can be realized at one common stage of a filtered diagram.
A local state classifier of is a colimit of the inclusion of the wide subcategory containing all objects and only monomorphisms into .
The Karoubi envelope freely splits idempotents. Its objects are pairs and a morphism is a map satisfying .
A regular category has finite limits, image factorizations into regular epimorphisms followed by monomorphisms, and pullback-stable regular epimorphisms.
For a regular category , the category has the objects of and subobjects of as relations . Composition takes the image of the pullback expressing existential quantification over the middle object.
The graph of is the relation represented by . Its converse is right adjoint to it in the pointwise order on relations.
For , the unit and counit are the natural transformations corresponding to identity morphisms under the adjunction. They satisfy and .
For , the right adjoint is full and faithful exactly when the counit is an isomorphism. Dually, is full and faithful exactly when the unit is an isomorphism.
The right Kan extension along is the right adjoint to precomposition by between suitable functor categories.
The colimit form of the Special adjoint functor theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family to a locally small category has a right adjoint.
A category with finite products is cartesian closed when every product functor has a right adjoint , called exponentiation by .
The category has metric spaces as objects and maps satisfying as morphisms. Its binary product uses the maximum metric.
The quotient metric is the largest metric on a quotient set for which the quotient map is non-expansive. It is obtained by taking infima of lengths of chains that may jump freely within equivalence classes.
A functor is final when every comma category is nonempty and connected. Colimits are unchanged after restriction along a final functor.
The Eilenberg-Moore category consists of algebras for a monad and their structure-preserving morphisms.
When each comparison functor has a left adjoint, iterating the Eilenberg-Moore comparison functor produces the monadic tower. Its monadic length is the least number of comparison steps needed to reach an equivalence; an equivalence has length zero and a non-equivalence that is already monadic has length one.
An adjunction is monadic when its comparison functor from the right-hand category to the Eilenberg-Moore category of the induced monad is an equivalence.
The precise monadicity theorem says that a right adjoint is monadic exactly when it reflects isomorphisms and creates coequalizers of the pairs whose images have split coequalizers.
A Lawvere theory is a small category with finite products generated by one object: every object is a finite power . A model in a finite-product category is a finite-product-preserving functor from the theory.
A monad on sets is finitary when its underlying functor preserves filtered colimits. Its operations on finite free algebras form a Lawvere theory.
For a frame , an -valued matrix is a function . Composition is matrix multiplication with join as addition and meet as multiplication.
An additive category is enriched in abelian groups, has a zero object and finite biproducts, and has bilinear composition.
An abelian category is an additive category with all kernels and cokernels in which every monomorphism and epimorphism is normal.
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Category theory is a branch of mathematics that focuses on the abstract study of mathematical structures and relationships between them. It provides a unifying framework to understand various mathematical concepts across different fields by focusing on the relationships (morphisms) between objects rather than the objects themselves. Here are some key concepts in category theory: 1. **Categories**: A category consists of objects and morphisms (arrows) that map between these objects. Each morphism has a source object and a target object.