**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 *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. |