Analytic Dictionary    By Zhaohua Luo (1997)    Let A be an analytic category.   analytic cover: a unipotent cover consisting of analytic monos.   analytic divisor: the divisor of analytic monos .   analytic mono: a coflat singular mono.   analytic topology: the framed topology determined by the analytic divisor..   atomic (or unisimple) object: a non-initial object such that any non-initial map to it is unipotent.   normal divisor: the divisor of normal monos   normal topology: the framed topology determined by the canonical divisor   coflat category: any map is coflat (or equivalently, any epi is stable).   coflat map: a map f: Y ® X is coflat if the pullback functor A/X ® A/Y along f preserves epis.   complement of a mono: a mono uc: Uc ® X is a complement of a mono u: U ® X if u and uc are disjoint, and any map v: T ® X such that u and v are disjoint factors through uc (uniquely). The complement uc of u, if exists, is uniquely determined up to isomorphism.   complement of a set of monos: a mono uc: Uc ® X is a complement of a set of monos {ui: Ui ® X} if each ui and uc are disjoint, and any map v: T ® X such that each ui and v are disjoint factors through uc (uniquely).   decidable category: an analytic category such that any strong mono is a direct mono.   decidable object: an object whose diagonal map is a direct mono.   direct cover: a unipotent cover consisting of direct monos   disjoint maps: two maps u: U ® X and v: V ® X are disjoint if the initial object 0 is the pullback of (u, v).   direct mono: an injection of a finite sum.   disjunctable category: any strong mono is a disjunctable mono.   disjunctable cogenerator: a set of cogenerators is called a set of disjunctable cogenerators if any object in the set is disjunctable.   disjunctable object: an object whose diagonal map is a disjunctable regular mono.   disjunctable strong mono: a strong mono with a coflat complement.   stable divisor: a class of maps containing isomorphisms which is closed under composition and stable under pullback.   extensive category: a category with finite stable disjoint sums.   extensive divisor: the divisor of direct monos   extensive topology: the framed topology determined by the extensive divisor   fraction: a subobject determined by a fractional mono.   fractional mono: a coflat and normal mono.   indecomposable component: a maximal indecomposable subobject.   indecomposable object: a non-initial object such that any non-initial map to it is indirect.   indirect map: a map which is not disjoint with any non-initial direct mono (or equivalently, not factors through any proper direct mono).   integral object: a non-initial reduced object such that any non-initial analytic subobject is epic.   lextensive category: an extensive category with finite limits.   local isomorphism: a map f: Y ® X such that, for any localization v: V ® Y, the composite f°v: V ® X is a localization.   local map: a map f: Y ® X which is not disjoint with any proper strong subobject of X.   local object: an object whose intersection of all the non- initial strong subobjects is a simple object.   locality: a fraction which is a local object.   localization: a fractional mono with a local object as domain (i.e. a mono which determines a locality).   localization at a prime: a locality which is the intersection of all the non-initial analytic subobjects that is not disjoint with a given prime.   localization at a simple subobject: a locality which is the intersection of all the analytic subobjects containing a given simple subobject.   locally decidable category: an analytic category such that any strong mono is locally direct.   locally decidable object: an object whose diagonal map is a locally direct mono.   locally direct mono: a mono which is an intersection of direct mono.   locally disjunctable category: any strong mono is locally disjunctable.   locally disjunctable mono: a strong mono which is an intersection of disjunctable strong monos.   locally disjunctable object: an object whose diagonal map is a locally disjunctable regular mono.   local-fractional-mono factorization: suppose f: Y ® X is a map. If f = g°l with l: Y ® Z a local map and f: Z ® X a fractional mono, then we say that (l, g) is a local-fractional factorization of f, and Z is the local image: of Y in X. It is easy to see that g is the intersection of all the fractions to X such that f factors, thus such a factorization is unique if exists.   normal mono: a map such that any of its pullback is not proper unipotent.   perfect category: any intersection of strong monos exists (or equivalent, the lattice of strong subobjects of any object is complete).   prime: an integral strong subobject.   radical: the unipotent reduced subobject of an object.   reduced category: any object is reduced.   reducible category: any non-initial object has a non-initial reduced subobject.   reduced model: the largest reduced strong subobject of an object.   reduced object: an object such that any unipotent map to it is epic.   regular mono: a map which can be written as the equalizer of some pair of maps.   residue: a simple fraction of a prime of an object.   semisingular map: a complement of a set of strong monos.   singular mono: a mono which is the complement of a strong mono.   simple object: a non-initial object such that any non-initial map to it is epic.   spatial category: any non-initial object has a prime.   strict analytic category: the Grothendieck topology determined by analytic covers is subcanonical (i.e. any representable presheaf is a sheaf).   subnormal divisor: a divisor whose maps are normal monos.   unipotent cover: a set of maps to an objects X such that any non-initial map to X is not disjoint with at least one map in the set.   unipotent map: a map such that any of its pullback is not proper initial.
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