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 u^c: U^c ® X is a complement of a mono u: U ® X if u and u^c are disjoint, and any map v: T ® X such that u and v are disjoint factors through u^c (uniquely). The complement u^c of u, if exists, is uniquely determined up to isomorphism.  

complement of a set of monos: a mono u^c: U^c ® X is a complement of a set of monos {u_i: U_i ® X} if each u_i and u^c are disjoint, and any map v: T ® X such that each u_i and v are disjoint factors through u^c (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.