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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 receives a unique morphism from every object.

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

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 $y>0$.

If there are injections $X\to Y$ and $Y\to X$, then there is a bijection $X\cong Y$.

Cardinal sums are cardinalities of disjoint unions, products are cardinalities of Cartesian products, and exponentials ${\kappa }^{\lambda }$ count functions from a set of size $\lambda$ to one of size $\kappa$.

An initial ordinal is an ordinal not equinumerous with any smaller ordinal. It is therefore the canonical ordinal representative of a cardinal.

An uncountable set is a set for which no bijection with a subset of the natural numbers exists.

A function $\Phi (q,Q)$ generates a canonical transformation through

A function $S(q,P)$ generates a canonical transformation through

For $N$ indistinguishable noninteracting classical particles, $Z_N=z_1^N/N!$, where $z_1$ is the normalized one-particle phase-space integral.