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A collection of arrows is a two-sided ideal when arbitrary precomposition and postcomposition preserve it. If a finite category has split idempotent morphisms, every ideal $I$ satisfying $I^2=I$ is generated by identities belonging to $I$. Factor an arrow $f\in I$ repeatedly as a longer product of arrows of $I$. Two prefixes coincide, since there are only finitely many arrows. The intervening endomorphism $t\in I$ fixes that prefix; an idempotent power $e$ of $t$ still fixes it. Write $e=ir$, $ri=1_B$. Then $1_B=rei\in I$, and $f$ factors through $B$. Applied to the least covering sieves of a finite site, this proves rigid coverage.

An image of $f:A\to B$ is the least subobject $m:I\hookrightarrow B$ through which $f$ factors. In a category with pullback in a category constructions this gives a strong epimorphism–monomorphism factorization $f=me$: pull back any mono in a lifting square for $e$ to $I$; 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.

An isomorphism of categories is a functor with a strictly inverse functor. It is equivalently bijective on objects and on each hom-set. An equivalence of categories only requires inverse composites up to invertible natural transformations. For example, the category of partial functions and the category of pointed sets are equivalent but their actual object collections prevent an isomorphism: the former has one zero object, the empty set, while the latter has distinct singleton pointed set objects that are all zero objects.

For a right-exact additive functor out of an abelian category with enough projectives, apply the functor to a projective resolution and take homology objects. The zeroth derived functor is $F$, positive derived functors vanish on projective objects, and the degreewise split horseshoe lemma produces a long exact sequence from every short exact sequence in an abelian category. The target can be any abelian category, including an opposite category of one.

A locale is an object of the opposite of the category of frames. A map $f:X\to Y$ is specified by a frame homomorphism $f^{\ast }:\mathcal{O}(Y)\to \mathcal{O}(X)$. This makes open-set algebra primary and allows spaces without enough ordinary points. A point of a frame is the inverse-image map of a point of a locale.

A specification of the meaning of a mathematical term or notation. A structural definition specifies conditions, such as the group axioms; a definition by a universal property characterizes an object by its arrows. Such a characterization does not by itself prove that the object exists. For example, existence of an exponential object is an additional property of a category.

A complete and cocomplete category with designated model weak equivalences, model fibrations and model cofibrations, closed under retracts. Weak equivalences satisfy two-out-of-three; maps admit the two cofibration/fibration factorizations, and either acyclic class has the required lifting property against the opposite class. These axioms organize homotopy-theoretic calculations abstractly.

A morphism of monads on one category is a natural transformation $\alpha :S\Rightarrow T$ with $\alpha {\eta }^S={\eta }^T$ and $\alpha {\mu }^S={\mu }^T(T\alpha )(\alpha S)$. These equations preserve the unit and multiplication. A terminal endofunctor in a full composition-closed subcategory containing the identity receives a unique such morphism from every monad whose functor belongs to that subcategory.

A category equipped with a monoidal tensor product, a monoidal unit object, and natural associators and unitors satisfying the pentagon and triangle axioms.

A categorical morphism is an arrow with a specified source and target in a category. An arrow $f:A\to B$ composes with $g:B\to C$ to give $gf:A\to C$. Composition is associative, and every object has an identity arrow acting as a unit. A functor preserves the sources, targets, identities and compositions of these arrows. In a locally small category, the arrows between fixed objects form a hom-set.

A natural transformation between functors $F,G:\mathcal{C}\to \mathcal{D}$ assigns a categorical morphism ${\alpha }_A:F(A)\to G(A)$ to every object of a category $A$, withThis naturality condition says that transforming the input before transporting it gives the same result as transporting it first and transforming the output. It constrains the whole family of arrows, so unrelated choices of a map for each object generally fail it.

For a concrete example, compare the functor that leaves every set and function unchanged with the power set functor using images of subsets. The maps ${\eta }_X:X\to \mathcal{P}(X)$ given by $x\mapsto \{x\}$ form a natural transformation: for any function $f:X\to Y$,The right-hand panel of the category illustration checks the square at $x=3$ with $f(x)=x+1$: both routes end at $\{4\}$. The statement works for every function, rather than depending on a preferred labeling of a particular set.

Map two disjoint walking arrows to the two mutually inverse arrows of the walking isomorphism. Their images generate the entire target, so the functor is an extremal epimorphism. Its kernel pair only identifies the matching endpoint objects, not the inverse-composition relations. Mapping both arrows to a nonidentity idempotent in a one-object category equalizes that kernel pair but cannot factor through the target groupoid. Thus the functor is not a regular epimorphism, although the category of small categories has finite limits.

The objects are the entities between which a category specifies morphisms. Each object has an identity morphism.

Use the preservation obstructions to extending an adjoint chain: a functor having a further left adjoint must preserve categorical limits, while one having a further right adjoint must preserve colimits.

At the left end of the presheaf category chain, $\Lambda$ does not preserve the terminal object. The terminal categorical presheaf has singleton value everywhere, whereas $(\Lambda 1)(S)=\varnothing$ because $S\ne \varnothing$. Thus $\Lambda$ cannot itself be a right adjoint. At the right end, $\nabla$ does not preserve the initial object: $(\nabla \varnothing )(\varnothing )=\{\ast \}$ since the empty open set is proper. Thus $\nabla$ cannot itself be a left adjoint.

For the category of small categories chain, take two functors from the terminal category $1$ to the two-object indiscrete category $I\{0,1\}$, selecting its distinct objects. Their equalizer is the empty category. But their images under $C={\pi }_0$ are the same function between singleton sets, whose equalizer is a singleton. Hence $C$ does not preserve equalizers and has no further left adjoint. Finally, $I1⨿I1$ is the two-object discrete category, while $I(1⨿1)$ has arrows between its distinct objects. The canonical comparison is not an isomorphism of categories, so $I$ fails to preserve this coproduct and has no further right adjoint.

A monoidal category consists of a category $\mathcal{C}$, a bifunctor $\otimes :\mathcal{C}\times \mathcal{C}\to \mathcal{C}$, a monoidal unit object $I$, and natural isomorphismsThe associator obeys the pentagon identity for a monoidal category,and the unitors obey the triangle identity for a monoidal category,Composition in a category here acts from right to left. The first equation compares two paths from $((A\otimes B)\otimes C)\otimes D$ to $A\otimes (B\otimes (C\otimes D))$; the second compares two ways of removing the intervening unit.

It is precisely a bicategory with one object $\ast$. The category $\mathcal{B}(\ast ,\ast )$ is $\mathcal{C}$: its objects are the 1-morphisms and its morphisms are the 2-morphisms. Horizontal composition in a category is $\otimes$, the identity 1-morphism is $I$, and the associator and unitors give the weak associativity and unit constraints. Vertical composition in a category is ordinary composition in a category in $\mathcal{C}$; the interchange law follows because $\otimes$ is a bifunctor. The bicategory pentagon and triangle axioms are exactly the two equations above. Conversely the sole hom-category of any one-object bicategory gives these data, with the convention that horizontal composition in a category uses the indicated ordering of tensor factors.

Here is a precise base change of a pullback square statement. Suppose $A=B{\times }_{D}C$ is a pullback in a category and $t:D^{\prime }\to D$ is a morphism. Assume the pullbacks in a categoryexist, where $A\to D$ is the common composite. The induced square with vertices $A^{\prime },B^{\prime },C^{\prime },D^{\prime }$ is a pullback in a category; equivalently,Indeed, compatible morphisms $T\to B^{\prime }$ and $T\to C^{\prime }$ give morphisms $b:T\to B$, $c:T\to C$, and a common $v:T\to D^{\prime }$, with the composites of $b,c$ to $D$ both equal to $tv$. The original pullback in a category gives a unique morphism $h:T\to A$ with projections $b,c$. Its composite to $D$ is $tv$, so the defining pullback in a category for $A^{\prime }$ gives a unique morphism $T\to A^{\prime }$ above $h,v$. Its maps to $B^{\prime }$ and $C^{\prime }$ are the prescribed ones by uniqueness in those pullbacks in a category. Conversely, these projections recover $b,c,v$, so every possible factorization is the same. This proves the universal property without requiring arbitrary categorical limits in the ambient category.

For a functor $D:\mathcal{J}\to \mathcal{C}$ with finite indexing category, form the finite products in a categoryDefine $s,t:P\rightrightarrows Q$ by requiring the coordinate indexed by $u:i\to j$ to be $D(u){\pi }_i$ for $s$ and ${\pi }_j$ for $t$. Let $e:L\to P$ be their equalizer, and put ${\lambda }_j={\pi }_{j}e$. The equality $se=te$ says exactly that $D(u){\lambda }_i={\lambda }_j$ for every morphism $u$, so these morphisms form a categorical cone.

For any other categorical cone $(x_j:X\to D(j))$, the product in a category gives a unique $x:X\to P$ with ${\pi }_{j}x=x_j$. The categorical cone equations give $sx=tx$, so the equalizer gives a unique $\bar{x}:X\to L$ with $e\bar{x}=x$. This is exactly the universal property of a categorical limit. Empty indexing products are terminal objects, so the construction includes the empty diagram in a category. Thus finite products and equalizers give all finite limits.

A precise commutation of iterated categorical limits statement is the following. For small categories $\mathcal{I},\mathcal{J}$ and $F:\mathcal{I}\times \mathcal{J}\to \mathcal{D}$, if every column categorical limit and the resulting $\mathcal{J}$-limit exist, thenIf every row categorical limit and the resulting $\mathcal{I}$-limit also exist, thenEach isomorphism is the unique one identifying all projections to $F(i,j)$. Indeed, the preceding part proves that each iterated categorical limit has the universal property of the whole product diagram in a category. Its uniqueness gives the canonical isomorphism. Equivalently, categorical cones with vertex $X$ over the product diagram in a category are exactly compatible families, indexed by $j$, of categorical cones over $F(-,j)$; regrouping the same compatibility equations cannot change their representing object. In a complete category all the stated existence hypotheses hold for small $\mathcal{I},\mathcal{J}$. The theorem is about categorical limits with categorical limits, and makes no unconditional claim about commuting a categorical limit with a colimit.

A poset $P$ becomes a category with the elements of $P$ as objects and exactly one morphism $x\to y$ when $x\le y$, with no morphism otherwise. Reflexivity supplies the identity morphisms and transitivity supplies composition in a category; uniqueness of morphisms proves associativity. A functor $f:P\to Q$ is exactly a monotone function: it must send $x\le y$ to $f(x)\le f(y)$, and this condition suffices because all the required morphisms are unique. Antisymmetry says that isomorphic objects of this category are equal.