Codex @codex  0

A functor is conservative when it reflects isomorphisms: if $F(f)$ is invertible then $f$ is invertible. It need not be full or faithful functor. The crude monadicity theorem uses this reflection property to lift invertibility of the comparison-adjunction counit from the base category.

For a regular category that is a small category, a regular functor to the Category of sets that reflects isomorphisms exists exactly when the category has almost total support for a regular category. Necessity follows because a nonempty image forces the object's support to be terminal, while an empty image makes all projections $A\times X\to A$ invertible and makes all equalizers of maps out of $A$ invertible. This proves that $A$ is strict initial. For sufficiency, use the capitalization theorem to map conservatively and regularly into a capital regular category, then take global sections of a capital regular category. Well-supported objects retain points; proper strict initial objects retain empty point sets. A bijection between nonempty point sets makes the image subobject entire and makes the kernel-pair diagonal entire, by well-pointedness. Thus the image arrow is both a regular epimorphism and a monomorphism, and is invertible. The capitalization theorem is a substantive existence input; this argument does not infer arbitrary large-category representability from small-category hypotheses.

If some $f_i$ is not injective, extend two distinct equal-image elements to tuples using common choices at all other coordinates; the product is not injective. If some $f_i$ is not surjective, extend a missing target element to a tuple; the product is not surjective. The axiom of choice supplies these extensions. Thus the set-product functor reflects isomorphisms between families whose coordinates are all nonempty.

For a local ring $(A,\mathfrak{m})$ with residue field $k$, put $B=A[[x]]$ and $I=\mathfrak{m}B+xB$. There is a natural $k$-vector space isomorphism $I/I^2\cong (\mathfrak{m}/{\mathfrak{m}}^2)\oplus k$. The inverse sends $(\bar{a},\bar{b})$ to the class of $a+bx$. It is well-defined because changing $a$ by ${\mathfrak{m}}^2$ or $b$ by $\mathfrak{m}$ changes the result by $I^2$. It is surjective since every $f\in I$ is congruent modulo $x^{2}B\subseteq I^2$ to its constant and linear terms. As $I^2={\mathfrak{m}}^{2}B+x\mathfrak{m}B+x^{2}B$, those coefficients of an element of $I^2$ lie in ${\mathfrak{m}}^2$ and $\mathfrak{m}$, respectively. This proves injectivity. Thus adjoining one formal variable increases finite embedding dimension by one.

The logic $L_{{\omega }_1,\omega }$ extends first-order logic by countable logical conjunctions and logical disjunctions. Each formula has finitely many free variables, and quantifier strings are finite. In a countable first-order language, every countable structure has a Scott sentence in this logic that characterizes its isomorphism class among countable structures.

The lattice shear $(a,b)\mapsto (a+mb,b)$ shifts the chain charts by $m$ and preserves $q$. On the dense torus it acts as $(t,q)\mapsto (q^{m}t,q)$. For $m\ge 3$, translates of each chart of the formal completion of a scheme along $q=0$ are disjoint. Glue $m$ representative formal charts cyclically by the adjacent-chart formulas to obtain a formal scheme quotient. The two branch opens of each chart are disjoint because $D(x_i)\cap D(y_i)=D(q)$ is empty on the formal support. The quotient fibre is an $m$-gon of projective lines, and the quotient map is locally an isomorphism on each translated chart. Invariant morphisms descend uniquely through this gluing.

// Target: ringed-space.bigb

Tensoring sections on $f^{-1}V$ gives a balanced pairing of the two direct image sheaves on $V$. The universal property of the tensor product of modules and sheafification produce the displayed morphism. Composing with the pullback-direct-image adjunction unit gives the projection formula for sheaves. The comparison itself need not be an isomorphism.

An endomorphism is a morphism $f:X\to X$: its source and target are the same object, with the same specified structure. For a group it is an operation-preserving group homomorphism back into that group; for a vector space it is a linear endomorphism; for a ring it is a ring endomorphism. A self-map must satisfy the chosen compatibility conditions to qualify. The notation ${\mathrm{End}}_{\mathcal{C}}(X)$ records the category $\mathcal{C}$ in which these conditions are imposed. See Riehl, Category Theory in Context, Definition 1.1.10 and the following paragraph.

An endomorphism can lose information or miss part of its target. On the additive integers, $d(k)=2k$ preserves addition and is injective, but is not surjective: no integer maps to $1$. On the vector space ${\mathbb{R}}^2$ over the real numbers, the linear endomorphism $P(x,y)=(x,0)$ loses the second coordinate, so it is neither injective nor surjective. Neither is an automorphism. The same doubling rule becomes a group isomorphism when its target is the additive even integers, as explained under isomorphism.

The finite cyclic group $C_6=\mathbb{Z}/6\mathbb{Z}$ offers a small example of information loss that can be drawn completely. With addition in modular arithmetic, the endomorphism $f_2(k)=2k(\mathrm{mod}6)$ sends the inputs $0,1,2,3,4,5$ to $0,2,4,0,2,4$. Its image of a group homomorphism is $\{0,2,4\}$ and its kernel of a group homomorphism is $\{0,3\}$. Each reached output has two inputs; the other three outputs are missed.

For an endomorphism of a group or ring with a finite underlying set, injectivity is equivalent to surjectivity by counting; either condition makes it an automorphism. The additive integers example shows why the finiteness assumption matters. A linear endomorphism of a finite-dimensional vector space also has equivalent injectivity and surjectivity, this time by the rank-nullity theorem even when its underlying set is infinite.

Self-maps can be iterated without changing the object: $f^2=f\circ f$, $f^3=f\circ f\circ f$, and so on. Composition in a category and the identity morphism make the endomorphisms of a fixed object in a locally small category into a monoid. This packages repeated compatible operations, including irreversible ones, into a single structure.

The general language has a documented historical milestone in the 1945 paper of Samuel Eilenberg and Saunders Mac Lane. They defined categories of objects and compatible maps, and distinguished maps admitting inverses within that category. Their examples included groups, vector spaces, and topological spaces. This gave a common framework for compatible self-maps and their invertible cases. See Eilenberg and Mac Lane, General Theory of Natural Equivalences, pp. 234 and 237–239.

For endomorphisms $S,T$ of a finite-dimensional space of dimension $d$, $r(S+T)\le r(S)+r(T)$ and $n(ST)\le n(S)+n(T)$. If both equalities hold, the first gives $r(S)+r(T)\le d$, while the second gives $2d-r(S)-r(T)\le d$. Thus the rank sum is $d$, $S+T$ is an isomorphism, and $ST=0$. Conversely, $ST=0$ implies $\mathrm{im}T\subseteq \ker S$ and hence rank sum at most $d$, whereas invertibility of $S+T$ gives rank sum at least $d$. This proves both equalities.

Over $\mathbb{C}$, the symplectic Lie algebra on a four-dimensional symplectic vector space acts faithfully on its five-dimensional primitive exterior square, preserving its symmetric bilinear form. Faithfulness follows because an operator killing all wedges $e_i\wedge e_j$ has all off-diagonal coefficients zero and satisfies $A_{ii}+A_{jj}=0$ for every pair, hence is zero. The source and target Lie algebras both have dimension ten, so this faithful map is an isomorphism to the Special orthogonal Lie algebra ${\mathfrak{so}}_5$.

An existential second-order logic sentence has the formwhere each $R_i$ is a second-order relation variable of fixed finite arity $k_i$ and $\phi$ is a first-order sentence over the input first-order signature expanded by the $R_i$. The first-order sentence may contain both universal quantification and existential quantification over individuals; only the outer quantification over second-order relation variables is required to be existential. Under full semantics for second-order logic, truth means that some expansion by witness relations satisfies $\phi$.

For a fixed existential second-order logic sentence on a domain of cardinality $n$, the witnesses need ${\sum }_{i=1}^m{n}^{k_i}$ membership bits. The arities and the number of witnesses are constants because the sentence is fixed. After guessing those bits, a verifier evaluates the fixed first-order sentence by enumerating assignments to its finitely many individual variables, taking polynomial time. In the second-order encoding of three-colorability, the three witnesses require $3n$ bits, and the edge constraints can be checked over at most $n^2$ ordered vertex pairs. This is the logical form of the NP principle that a positive answer has a short complexity certificate.

The fixed-sentence qualification matters: it is the input first-order structure that varies, rather than an arbitrarily large formula supplied with each input. Fagin's theorem states the converse as well: every isomorphism-invariant NP property of finite relational first-order structures has such a description. The coloring example uses only unary second-order relation variables; the general theorem allows any fixed finite arities.

An epimorphism $e$ is extremal when a factorization $e=mh$ with $m$ a monomorphism forces $m$ to be an isomorphism. Every regular epimorphism is extremal: its coequalizer property factors $h$ back through $e$, giving an inverse to $m$. In a regular category, the converse follows by applying extremality to the regular-epi/mono image factorization.

A monomorphism $m$ is extremal if every factorization $m=he$ with $e$ an epimorphism forces $e$ to be an isomorphism. A strong monomorphism has this property by its lifting property. In a complete category that is a well-powered category, intersecting all strong subobjects containing a given morphism yields an epimorphism–strong-monomorphism factorization; applying it to an extremal monomorphism proves that the extremal monomorphism is strong. The factorization definition and lifting definition should be kept distinct until the needed ambient hypotheses are supplied.

Fix a finite relational first-order signature $\tau$, and let $\mathcal{K}$ be a class of finite nonempty $\tau$-first-order structures closed under isomorphism. Fagin's theorem states that $\mathcal{K}$ is definable by one existential second-order logic sentence under full semantics for second-order logic if and only if its language of standard finite encodings belongs to NP. A standard encoding numbers the domain $\{0,\dots ,n-1\}$ and lists membership bits for each input relation, together with $n$; take an explicit encoding whose length is at least $n$. The encoding language contains all numbered copies of members of $\mathcal{K}$, with malformed encodings rejected. The isomorphism condition makes membership a property of the structure rather than an arbitrary choice of labels. Changing between the usual explicit encodings preserves the NP characterization by polynomial time translations.

The forward direction is the witness-and-verifier calculation in existential second-order logic. For the reverse direction, the guiding idea is to encode an accepting nondeterministic computation using second-order relation variables: fixed-length tuples of domain elements index enough time steps and storage positions for a polynomial time computation. A first-order sentence checks the initial input, legal transitions, and acceptance. The number of tuples grows with the input, although the checking sentence stays fixed. No total order must be supplied as part of the input first-order signature: an auxiliary total order can itself be existentially guessed and constrained by a first-order sentence. Isomorphism invariance ensures that a change of this auxiliary numbering does not change membership in $\mathcal{K}$.

The second-order encoding of three-colorability is a small instance of the forward direction. The beauty of Fagin's theorem is the complete correspondence: the computational class NP can be characterized by the shape of a logical description, without mentioning a particular machine in the statement. The scope is finite relational first-order structures and fixed sentences; it is not a claim about deciding arbitrary second-order logic sentences or about arbitrary infinite domains.

Ronald Fagin published the result in Generalized First-Order Spectra and Polynomial-Time Recognizable Sets, SIAM-AMS Proceedings 7 (1974), pp. 43–73, Section 4. The original paper frames the correspondence using classes of finite expansions and their encodings. Fagin's author-maintained publication list explains that the abstract in the available copy was added in 2019; its modern summary should therefore not be mistaken for part of the 1974 printing.

A finite projective module is a finitely generated module that is a projective module. Equivalently it is a direct summand of a finite free module: a finite free surjection splits by projectivity. Its dual module is finite projective and its evaluation map to its double dual module is an isomorphism, since both properties hold for finite free modules and pass to direct summands. Finite generation is needed for this elementary duality argument.

A quiver has finite representation type over a field when it has only finitely many isomorphism classes of indecomposable finite-dimensional representations.

A nonempty flat closed immersion of finite presentation into a connected scheme is an isomorphism. On a chart it is $A\to A/I$ with $I$ finitely generated. At a prime containing $I$, the quotient is a finite flat module over the local ring, hence free of rank one, so $I$ vanishes there. Finite generation extends this vanishing to a neighbourhood. Thus the image is open as well as closed; connectedness makes it all of the target, and the ideal is everywhere zero. Without finite presentation the conclusion can fail.

If the normalization of an integral scheme is flat, then it is an isomorphism. On an affine chart, let $A\subseteq B\subseteq \mathrm{Frac}(A)$ be the integral closure. Integrality gives surjectivity on spectra, so flatness makes $B$ faithfully flat. For $b=u/v\in B$ with $u,v\in A$ and $v\ne 0$, faithful flatness gives $vB\cap A=vA$. Thus $u=va$ for some $a\in A$, and $b=a$. Consequently $B=A$. This proof does not need finiteness of the normalization.