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For $0<s<a$,

For $a>0$ and real $c$, integrate $e^{-a{z}^2}$ around a wide rectangle between the real axis and $\mathrm{Im}z=c$. The vertical contributions vanish as the width tends to infinity, so

If $f$ is analytic and bounded by $K$ on $\mathrm{Re}z>0$, then Cauchy's derivative estimate giveson $\mathrm{Re}z>c$. Integrating along line segments gives a Lipschitz bound with the same constant in that smaller half-plane.

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

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.

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.

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 $\mathcal{C}$.

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

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 $ri=1_B$.

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

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 terminal object $1$ of a category receives exactly one morphism $X\to 1$ from every object $X$. Any two terminal objects are uniquely isomorphic by composing their unique connecting maps.

A singleton is terminal in the Category of sets, and the trivial group is terminal among groups. The empty set is not terminal, because no function from a nonempty set to it exists. An object that is both initial and terminal is a zero object; this occurs for the trivial group but not in the category of all sets.

A subobject of an object $A$ in a category is an equivalence class of monomorphisms $m:B\to A$. Two such maps are equivalent if an isomorphism between their domains makes the maps into $A$ agree. This removes dependence on the name of the chosen domain.

In the Category of sets, subobjects correspond to subsets; in the category of groups, they correspond to subgroups. The map into $A$ remains essential: two abstractly isomorphic subgroups can define different subobjects if they occupy different positions in $A$. Dually, quotient objects are equivalence classes of epimorphisms out of an object.

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.

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

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

A functor assigns an object of a category $F(A)$ to each object $A$ and a categorical morphism $F(f):F(A)\to F(B)$ to each arrow $f:A\to B$. It preserves the rules for composing arrows:For example, the forgetful functor from the category of groups to the Category of sets keeps the underlying sets and maps of elements while forgetting the group operations. A group homomorphism remains a function, and composing the underlying functions gives the underlying map of the composite group homomorphism.

Another example sends a set $X$ to its power set $\mathcal{P}(X)$. It sends a function $f:X\to Y$ to the function $\mathcal{P}(f):\mathcal{P}(X)\to \mathcal{P}(Y)$ defined by $S\mapsto f(S)$, the image of the subset $S$. For $f(x)=x+1$ and $g(x)=2x$, the subset $\{0,2\}$ goes to $\{1,3\}$ and then $\{2,6\}$; applying the composite $g\circ f$ directly also gives $\{2,6\}$. The equality of these routes expresses preservation of composition in a category.

The Cartier class group is the group of Cartier divisors modulo principal Cartier divisors. On an integral scheme it is naturally $H^1(X,{\mathcal{O}}_X^{\ast })$.

For a Cartier divisor $D$, the associated line bundle $\mathcal{O}(D)$ has local sections given by meromorphic functions $f$ satisfying $(f)+D\ge 0$. Its canonical meromorphic section has divisor $D$.

The positive quadrant of the Cartesian plane is the set of points $(x,y)$ with $x>0$ and $y>0$.