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A monoidal category with natural invertible braidings $c_{X,Y}:X\otimes Y\to Y\otimes X$ satisfying the two hexagon axioms. The equation $c_{Y,X}{c}_{X,Y}=1$ is an additional symmetric condition, not a braided axiom.

An object with multiplication $m:A\otimes A\to A$ and unit $j:I\to A$ satisfying associativity and unit diagrams using the ambient associators and unitors.

The categorical dual of a monoid object: an object with coassociative comultiplication and a counit, with the ambient constraints included.

Using the convention $-\otimes M⊣[M,-]$, every tensoring-on-the-right functor has a right adjoint functor. Evaluation is $[M,N]\otimes M\to N$. Naming left and right closure varies in the literature, so the adjunction fixes the convention.

Objects $X,Y$ with an evaluation morphism $e:X\otimes Y\to I$ and a coevaluation morphism $n:I\to Y\otimes X$ satisfying both snake identities. These are categorical duality data, distinct from the locally convex dual pair.

An invertible morphism $y:X\otimes X\to X\otimes X$ satisfying $(y\otimes 1)(1\otimes y)(y\otimes 1)=(1\otimes y)(y\otimes 1)(1\otimes y)$ with coherent parentheses. It produces representations of the braid groups; a self-braiding is an example.

If $f(T)$ is monic of degree $k$ over a commutative ring, every polynomial has a unique expression $g=qf+r$ with $\mathrm{deg}r<k$. Cancel the highest remaining term successively; the leading coefficient one needs no inversion. Uniqueness follows because multiplication by a monic polynomial raises the degree of every nonzero polynomial by $k$, even when the ring has zero divisors. The same argument applies to an even central variable and central coefficients in a graded commutative algebra.

A monad whose endofunctor is an opmonoidal functor and whose unit and multiplication are opmonoidal natural transformations. Its Eilenberg-Moore category has a tensor of algebras $(X,a)\otimes (Y,b)=(X\otimes Y,(a\otimes b)T_2)$ and unit $(I,T_0)$.

The natural maps $\eta :1\Rightarrow T$ and $\mu :T^2\Rightarrow T$ defining a monad, subject to the associativity and two unit identities.

On the Category of sets, the list monad sends $X$ to all finite ordered lists of elements of $X$, including the empty list. Its unit inserts a singleton list and its multiplication concatenates a list of lists. Functoriality is entrywise application of functions. Ordered flattening proves the monad laws. This is the free-monoid monad, so order must not be discarded.

A uniform solid sphere of mass $M$ and radius $a$ has the indicated moment of inertia about any diameter. Integrating the squared perpendicular distance against uniform mass density gives $I=8\pi \rho a^5/15=2M{a}^2/5$. It is different from the value for a thin spherical shell and determines the rotational contribution to kinetic energy during rolling without slipping.

The distributional equation $\Delta u=\mathrm{div}(bu)+cu+f$ becomes the smooth interior equation $\Delta u_{\sigma }=\mathrm{div}(bu{)}_{\sigma }+(cu{)}_{\sigma }+f_{\sigma }$. The products are convolved as wholes. Only coefficient bounds on compact interior sets are needed, and the divergence-forcing interior H1 estimate provides a uniform first derivative bound.

Mollification is the regularization of a function or distribution by convolution with a rescaled mollifier. It creates smooth functions on interior regions, preserves constant-coefficient distributional equations there, and converges back to the original object in the appropriate local topology as the radius tends to zero. The process is distinct from the kernel used to perform it.

Generating a potential with an acceptable stable minimum for previously undetermined compactification scalar fields. It can involve fluxes, branes and nonperturbative effects, with supersymmetry and consistency restrictions. A plausible phenomenological model must determine or suitably control these fields rather than leave arbitrary massless scalars and varying couplings.

A composition series of a module is a finite chain of submodules whose nonzero successive quotient modules are simple modules. Every finite-dimensional Lie algebra representation has such a chain, viewed as a module over the universal enveloping algebra.

For central elements, each multiplication map on the preceding quotient must be injective; a usual convention also requires the final quotient to be nonzero. The augmented Koszul complex on central ring elements tensored with $M$ is then a quasi-isomorphism to the final quotient in degree zero.

A ring is left hereditary if every submodule of a left projective module is projective. Equivalently, every left module has a projective resolution of length at most one. The standard projective resolution of a quiver representation, used also for arbitrary infinite-dimensional modules, proves that path algebras are left hereditary.

A finite-dimensional module $X$ degenerates to $Y$ when $Y$ has a representative in the closure of $X$'s change-of-basis orbit in the representation variety of an associative algebra. The full orbit of $Y$ then lies in that closure. The relation is transitive because orbit closures are closed and invariant under change of basis. A split extension as a degeneration shows that a module degenerates to the direct sum of its composition factors.

For a finite-index subgroup of the modular group, the quotient of the complex upper half-plane becomes a compact Riemann surface after adjoining its cusp classes. At an elliptic fixed point use the quotient coordinate for the finite stabilizer; at a cusp use its exponential local parameter. Weight-zero modular forms descend to holomorphic functions on this compact surface, and hence are constant by the maximum modulus principle.

The compact quotient for the Gamma 0 congruence subgroup at level two has projective index three, two modular cusps of cusp widths one and two, one order-two elliptic orbit and no order-three elliptic orbit. It has genus zero. It differs from the level-two principal-congruence curve for $\Gamma (2)$, which has three modular cusps.