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A geometric embedding is a geometric morphism whose direct-image functor is full and faithful. Its inverse image presents a left-exact reflective subcategory of the target topos. It is the second part of the surjection-embedding factorization of a geometric morphism.

For $f:\mathcal{E}\to \mathcal{F}$, the comonad $G=f^{\ast }{f}_{\ast }$ is Cartesian, and its coalgebra topos $\mathcal{D}$ gives a factorization $f=i\circ p$. Here $p^{\ast }$ is the faithful forgetful functor and $i^{\ast }$ is the comparison functor. Its right adjoint sends $(A,a)$ to $\mathrm{Eq}(f_{\ast }a,{\eta }_{f_{\ast }A})$; applying $f^{\ast }$ identifies the comparison counit with the equalizer $a:A\to GA$ of $Ga,{\delta }_A$, proving that $i_{\ast }$ is full and faithful.

A geometric morphism is essential when its inverse image has an additional left adjoint. For a functor of small categories, precomposition on presheaves has left Kan extension and Right Kan extension as its two adjoints. Precomposition preserves finite limits pointwise, so this is an essential geometric morphism.

The inverse image is a finite-limit-preserving left adjoint with right adjoint $f_{\ast }$. It preserves arbitrary colimits, and hence images as well as finite limits. These properties preserve the interpretation of geometric formulas. It need not preserve Heyting implication or every universal quantifier.

Every Grothendieck topos has a unique geometric morphism to sets, up to isomorphism. Its inverse image is $\Delta S={\coprod }_{s\in S}1$ and its direct image is $\Gamma A=\mathrm{Hom}(1,A)$. A geometric inverse image out of sets must have this form because it preserves coproducts and the terminal object. A local topos is one for which $\Gamma$ itself is an inverse image.

Geometric morphisms into a sheaf topos correspond to continuous flat functors from its site into the domain topos, with continuity meaning that covers become jointly epimorphic families. Pulling back sheafified representables gives the functor; the tensor construction gives the inverse image in the other direction. This representation theorem should not be confused with the separately named Diaconescu theorem about choice and excluded middle.

A geometric formula is built from atoms by finite conjunction, arbitrary small disjunction and existential quantification. Empty conjunction and disjunction give truth and falsity. In a Grothendieck topos, these constructions use finite limits, unions and images. Formula contexts have finitely many variables.

A set of sequents between geometric formulas over a many-sorted signature. Its geometric syntactic category and geometric syntactic topology yield its classifying topos. A model satisfies a sequent when the interpreted antecedent subobject is contained in the consequent in the indicated context.

A nonconstant geodesic whose position and tangent velocity repeat after a positive time. On a flat torus it is a straight line modulo the lattice, closing when its displacement is a Euclidean lattice vector. A primitive closed geodesic traverses its image without being an iterate of a shorter periodic geodesic.

A filter is generic over its ground model exactly when it meets each ground-model maximal forcing antichain in one point. Extensions of a maximal antichain form a dense set, while any ground dense set contains an antichain maximal in the whole forcing. Directedness permits at most one antichain member in a filter.

For a proper subgroup $H$ of any group $G$, its complement is a generating set of a group. Choose $x\notin H$. For every $h\in H$, also $xh\notin H$, and $h=x^{-1}(xh)$ lies in the subgroup generated by the complement. That subgroup consequently contains both $H$ and its complement. No finiteness assumption is needed.

For prime $p$, every order-$p$ element of $G{L}_2({\mathbb{F}}_p)$ is conjugate to the displayed nonidentity unipotent matrix. Its cyclic subgroup has vector group orbits of sizes one or $p$; counting all $p^2$ vectors implies at least $p$ fixed vectors. Choose a nonzero fixed vector as first basis vector. The matrix becomes triangular with first diagonal entry one; its other diagonal entry satisfies $d^p=1$ and hence $d=1$ in the prime finite field. The off-diagonal entry is nonzero and can be rescaled to one. All matrices involved have invertible changes of basis.

For a character of modulus $p^k$, a primitive Dirichlet character attains the bound by finite Plancherel theorem. An imprimitive character with $k\ge 2$ has periodic values modulo $p^{k-1}$, so its frequency-one sum vanishes. The principal Dirichlet character modulo $p$ has unnormalized sum $-1$. The sign convention of the exponential does not affect its magnitude.

With gauge covariant derivative $D=d+A$, transforming the zero connection by a smooth group-valued map $g$ gives $A=-dg{g}^{-1}$. This potential has zero gauge curvature. A flat potential is locally of this form; global reconstruction can also depend on topology and holonomy. The sign changes when the definition of the covariant derivative changes.

Write $B_i=({\partial }_{i}g)g^{-1}$ and use $D=d+A$. The right Maurer-Cartan equation gives ${\partial }_x{B}_y-{\partial }_y{B}_x=[B_x,B_y]$. Thus $A_i=\alpha B_i$ has the displayed gauge curvature. Both $\alpha =0$ and $\alpha =-1$ are universally flat; the latter is a pure gauge potential. Commuting $B_x,B_y$ give additional flat cases. For the opposite convention $D=d-A$, the corresponding component formula instead has $\alpha (1-\alpha )$.

For ${\mathcal{F}}_{\mu \nu }={\partial }_{\mu }{\mathcal{A}}_{\nu }-{\partial }_{\nu }{\mathcal{A}}_{\mu }+ig[{\mathcal{A}}_{\mu },{\mathcal{A}}_{\nu }]$, the CP transformation of a non-Abelian gauge connection gives ${\mathcal{F}}_{\mu \nu }^{CP}(x)=-P_{\mu }{}^{\alpha }{P}_{\nu }{}^{\beta }{\mathcal{F}}_{\alpha \beta }^T(x_P)$. The commutator term transforms with the same sign as the derivative terms because $[X^T,Y^T]=-[X,Y{]}^T$. A simple component check is $F_{0i}\mapsto +F_{0i}^T$ and $F_{ij}\mapsto -F_{ij}^T$, evaluated at the reflected point.

If a complex scalar field uses $D_{\mu }\varphi =({\partial }_{\mu }+ie{A}_{\mu })\varphi$, its conjugate uses $(D_{\mu }\varphi {)}^{\ast }=({\partial }_{\mu }-ie{A}_{\mu }){\varphi }^{\ast }$. Under $\varphi \mapsto e^{-ie\alpha }\varphi$ and $A\mapsto A+d\alpha$, the two derivatives transform with opposite phases. Their product is gauge invariant. Applying the same plus-charge operator to both factors leaves an unwanted term proportional to ${\partial }_{\mu }\alpha$ and fails gauge invariance.

For $\mathcal{L}=-\mathcal{N}{F}_{\mu \nu }^a{F}^{a\mu \nu }/4$ with $\mathcal{N}>0$, rescale ${\mathcal{A}}^a=\sqrt{\mathcal{N}}{A}^a$ and $g_c=g/\sqrt{\mathcal{N}}$. The gauge field strength then has canonical component normalization. Masses inferred from matter couplings must use $g_c$, not an unnormalized coefficient. A matrix-trace convention $\mathrm{Tr}(t^a{t}^b)=\mathcal{N}{\delta }^{ab}$ fixes this factor.

For an odd prime $p$, embed the elementary abelian group $(C_p{)}^3$ and the Heisenberg group over a prime field regularly in $S_{p^3}$. Every nonidentity element has order $p$ and hence regular permutation cycle type $p^{p^2}$. The two subgroups have equal intersection counts with every symmetric-group conjugacy class, so they are Gassmann equivalent, but they are not isomorphic because one is abelian and the other is not. Using a compact simply connected isometric cover with ambient deck group $S_{p^3}$ gives quotients with these nonisomorphic fundamental groups; Sunada theorem makes them isospectral while their topology differs.

For signature $+---$, use ${\gamma }^0=\left(\begin{array}{cc}0& I_2\\ I_2& 0\end{array}\right)$ and ${\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$, where ${\sigma }^i$ are Pauli matrices. Their Clifford algebra follows directly from the Pauli anticommutator. The chirality matrix is ${\gamma }^5=\mathrm{diag}(-I_2,I_2)$, separating the two chiral components of a Dirac spinor.