Category

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.

The functor category has functors $\mathcal{C}\to \mathcal{D}$ as objects and natural transformations as morphisms.

A covariant functor is representable when it is naturally isomorphic to $\mathcal{C}(A,-)$ for some object $A$; a contravariant functor is representable when it is naturally isomorphic to $\mathcal{C}(-,A)$.

The Yoneda lemma gives natural bijectionsand their contravariant analogues.

A functor $F:\mathcal{C}\to \mathcal{D}$ is full and faithful when every mapis a bijection.

A category is small when its objects and morphisms form sets.

The category of sets has sets as objects and functions as morphisms.

An epimorphism $e:A\to B$ is right-cancellative: $fe=ge$ implies $f=g$. In the Category of sets and in abelian groups, epimorphisms are precisely the surjective homomorphisms.

A subobject of $A$ is an isomorphism class of monomorphisms into $A$. A quotient object is dually an isomorphism class of epimorphisms out of $A$.

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 $P$ is projective when every morphism $P\to B$ lifts through every epimorphism $A\twoheadrightarrow B$. Equivalently, the functor $\mathcal{C}(P,-)$ preserves epimorphisms.

A terminal object receives a unique morphism from every object.

A limit of a diagram $D:\mathcal{J}\to \mathcal{C}$ is a terminal cone over $D$: every other cone factors through it uniquely.

A finite limit is a categorical limit whose indexing category has finitely many objects and morphisms.

A product of objects $(A_i)$ is a universal object equipped with projections to every $A_i$.

The equalizer of parallel arrows $f,g:A\rightrightarrows B$ is a universal arrow $e:E\to A$ satisfying $fe=ge$.

A colimit is a universal cocone under a diagram: every other cocone factors through it uniquely.

Limits of shape $\mathcal{I}$ commute with colimits of shape $\mathcal{J}$ when, for every $D:\mathcal{I}\times \mathcal{J}\to \mathcal{C}$, 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 $\mathcal{C}$ is a colimit of the inclusion of the wide subcategory containing all objects and only monomorphisms into $\mathcal{C}$.

An endomorphism $e:A\to A$ is idempotent when $e^2=e$. It splits when $e=ir$ for maps $r:A\to B$, $i:B\to A$ satisfying $ri=1_B$.

The Karoubi envelope freely splits idempotents. Its objects are pairs $(A,e)$ and a morphism $(A,e)\to (B,f)$ is a map $u:A\to B$ satisfying $u=fue$.

A regular category has finite limits, image factorizations into regular epimorphisms followed by monomorphisms, and pullback-stable regular epimorphisms.

For a regular category $\mathcal{C}$, the category $\mathbf{Rel}(\mathcal{C})$ has the objects of $\mathcal{C}$ and subobjects of $A\times B$ as relations $A⇝B$. Composition takes the image of the pullback expressing existential quantification over the middle object.

The graph of $f:A\to B$ is the relation represented by $(1_A,f):A\hookrightarrow A\times B$. Its converse is right adjoint to it in the pointwise order on relations.

For $F:\mathcal{C}\to \mathcal{D}$ and $B\in \mathcal{D}$, the objects of $(B\downarrow F)$ are arrows $B\to FA$; its morphisms are commuting triangles induced by arrows in $\mathcal{C}$.

An adjunction is a natural bijection $\mathcal{D}(LB,A)\cong \mathcal{C}(B,RA)$.

For $F⊣G$, the unit $\eta :1\to GF$ and counit $\epsilon :FG\to 1$ are the natural transformations corresponding to identity morphisms under the adjunction. They satisfy ${\epsilon }_{F}F\eta =1_F$ and $G\epsilon {\eta }_G=1_G$.

For $F⊣G$, the right adjoint $G$ is full and faithful exactly when the counit $FG\to 1$ is an isomorphism. Dually, $F$ is full and faithful exactly when the unit $1\to GF$ is an isomorphism.

The right Kan extension along $K:\mathcal{C}\to \mathcal{D}$ is the right adjoint to precomposition by $K$ 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 $-\times A$ has a right adjoint $(-{)}^A$, called exponentiation by $A$.

An object $A$ of a cartesian closed category is tiny when exponentiation $(-{)}^A$ itself has a right adjoint.

The category $\mathbf{Met}$ has metric spaces as objects and maps $f$ satisfying $d(fx,fy)\le d(x,y)$ as morphisms. Its binary product uses the maximum metric.

A non-expansive map between metric spaces is a function $f$ such that $d(fx,fy)\le d(x,y)$.

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 $F:\mathcal{C}\to \mathcal{D}$ is final when every comma category $(B\downarrow F)$ is nonempty and connected. Colimits are unchanged after restriction along a final functor.

A functor $F:\mathcal{C}\to \mathcal{D}$ is a discrete fibration when every arrow $B\to FA$ has a unique lift with codomain $A$.