Sub-object classifier

Sub-object classifiers are examined in the mathematical sub-area of category theory. It is a monomorphism , so that each sub-object appears in this way as a pullback of this monomorphism along a uniquely determined morphism. The basic idea comes from the category of sets , in which a subset with the associated characteristic function can be identified.

Sub-object classifier for sets

Let it be defined by, which is obviously a monomorphism in , the category of sets. Furthermore, let it be a set and a subset. ${\displaystyle true:\{0\}\rightarrow \{0,1\}}$ ${\displaystyle true(0):=1}$ ${\displaystyle {\mathcal {Set}}}$ ${\displaystyle C}$ ${\displaystyle D\subset C}$

${\displaystyle \chi _{D}:C\rightarrow \{0,1\},\quad c\mapsto {\begin{cases}1&{\text{ wenn }}c\in D\\0&{\text{ wenn }}c\notin D\end{cases}}}$

be the characteristic function of the subset . Denoting the inclusion with , one has the following diagram ${\displaystyle D\subset C}$ ${\displaystyle D\subset C}$ ${\displaystyle \iota }$

${\displaystyle {\begin{array}{ccc}D&\rightarrow &\{0\}\\{\Bigg \downarrow }\iota &&{\Bigg \downarrow }true\\C&{\xrightarrow[{}]{\chi _{D}}}&\{0,1\}\end{array}}}$,

which is obviously commutative, because both possible paths map everything to 1. What's more, this diagram is actually a pullback because the diagram does too ${\displaystyle f:E\rightarrow C}$

${\displaystyle {\begin{array}{ccc}E&\rightarrow &\{0\}\\{\Bigg \downarrow }f&&{\Bigg \downarrow }true\\C&{\xrightarrow[{}]{\chi _{D}}}&\{0,1\}\end{array}}}$

commutative, it must be, that is, there is an unambiguous factorization , whereby the mapping is with a restricted target set. Strictly speaking, one would also have to show the factoring of the arrow to , but since there can only be a single mapping of any set (one also says that it is a terminal object ), this is automatically fulfilled. Also, it's easy to imagine that this is the only figure that pulls the former chart back. Ultimately, this is nothing more than the usual identification of a subset with the associated characteristic function. These considerations motivate the following definition: ${\displaystyle f(E)\subset D}$ ${\displaystyle f=\iota \circ f|^{D}}$ ${\displaystyle f|^{D}}$ ${\displaystyle f}$ ${\displaystyle D}$ ${\displaystyle \{0\}}$ ${\displaystyle \{0\}}$ ${\displaystyle \{0\}}$ ${\displaystyle \chi _{D}}$ ${\displaystyle C\rightarrow \{0,1\}}$

definition

Let it be a category with a terminal object . A sub-object classifier is a monomorphism in , so that the following property is satisfied: For every monomorphism in there is exactly one morphism such that ${\displaystyle {\mathcal {C}}}$ ${\displaystyle 1}$ ${\displaystyle true:1\rightarrow \Omega }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle m:D\rightarrow C}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \chi _{m}:C\rightarrow \Omega }$

${\displaystyle {\begin{array}{ccc}D&\rightarrow &1\\{\Bigg \downarrow }m&&{\Bigg \downarrow }true\\C&{\xrightarrow[{}]{\chi _{m}}}&\Omega \end{array}}}$

is a pullback.

Examples

• According to the above, a sub-object classifier is in the category of sets. ${\displaystyle true:\{0\}\rightarrow \{0,1\}}$
• Let it be a monoid and the category of M spaces . A “legal ideal” in is a subset , so that applies to all . The empty set and are always legal ideals. The set of all legal ideals of is determined by the operation ${\displaystyle M}$ ${\displaystyle {\mathcal {B}}M}$ ${\displaystyle M}$ ${\displaystyle N\subset M}$ ${\displaystyle N\supset Na}$ ${\displaystyle a\in M}$ ${\displaystyle M}$ ${\displaystyle \Omega }$ ${\displaystyle M}$

${\displaystyle \Omega \times M\rightarrow M,\quad (N,a)\mapsto N\cdot a:=\{b\mid ab\in N\}}$

to a room. The one-element space with the trivial (and only possible) operation of is a terminal object in and ${\displaystyle M}$ ${\displaystyle 1=\{0\}}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {B}}M}$

${\displaystyle true:1=\{0\}\rightarrow \Omega ,\quad true(0):=M}$

is a sub-object classifier in . If there is a monomorphism, perform ${\displaystyle {\mathcal {B}}M}$ ${\displaystyle m:D\rightarrow C}$ ${\displaystyle M}$

${\displaystyle \chi _{m}:C\rightarrow \Omega ,\quad c\mapsto \chi _{m}(c):=\{a\in M\mid x\cdot a\in m(D)\}}$

the requested.

• Any small category can be known by the Yoneda embedding into the category of Prägarben on , that is the functor fully loyal to embed. Even if it doesn't have a sub-object classifier itself, there is always one in . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\hat {\mathcal {C}}}={\mathcal {Set}}^{{\mathcal {C}}^{op}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\hat {\mathcal {C}}}}$

Since a terminal object is in, it is easy to check that the functor marked with , which maps every object to 1 and all morphisms to , is a terminal object in . ${\displaystyle 1=\{0\}}$ ${\displaystyle {\mathcal {Set}}}$ ${\displaystyle {\hat {1}}}$ ${\displaystyle {\mathcal {C}}^{op}\rightarrow {\mathcal {Set}}}$ ${\displaystyle \mathrm {id} _{1}:1\rightarrow 1}$ ${\displaystyle {\hat {\mathcal {C}}}}$

We now define an object in , that is, a functor through ${\displaystyle \Omega }$ ${\displaystyle {\hat {\mathcal {C}}}}$ ${\displaystyle \Omega :{\mathcal {C}}^{op}\rightarrow {\mathcal {Set}}}$

${\displaystyle \Omega (C)}$= Set of all screens on ${\displaystyle C}$

and for a morphism ${\displaystyle f:C\rightarrow D}$

${\displaystyle \Omega (f):\Omega (D)\rightarrow \Omega (C),\quad \Omega (f)(S):=f^{*}(S):=\{h\mid f\circ h\in S\}}$

Designates the maximum sieve , i.e. the set of all morphisms with a goal , so is ${\displaystyle t(C)}$ ${\displaystyle C}$ ${\displaystyle C}$

${\displaystyle true:{\hat {1}}\rightarrow \Omega ,\quad true(C):{\hat {1}}(C)=\{0\}\rightarrow \Omega (C),\quad 0\mapsto t(C)}$

a sub-object classifier in . ${\displaystyle {\hat {\mathcal {C}}}}$

• Many other categories are equivalent to a pregroove category of the previous example and many subcategories contained therein, in particular categories of sheaves, have sub-object classifiers. The existence of a sub-object classifier is an integral part of the definition of a topos .
• The Abelian group category does not have a sub-object classifier. More generally, the category of links R modules over any ring does not have a sub-object classifier. ${\displaystyle {\mathcal {Ab}}}$ ${\displaystyle R}$

Representation of the sub-object functor

Sub-objects in a category are equivalence classes ( isomorphism classes ) of monomorphisms . We want to assume that the totality of the sub-objects of an object forms a set that we denote with . Furthermore, we want to assume that there are finite limits , especially pullbacks. If a morphism is in , there is a pullback for every subobject of that is represented by a monomorphism ${\displaystyle {\mathcal {C}}}$ ${\displaystyle m:S\rightarrow C}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle C}$ ${\displaystyle \mathrm {Sub} _{\mathcal {C}}(C)}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle f:C\rightarrow D}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle D}$ ${\displaystyle h:E\rightarrow D}$

${\displaystyle {\begin{array}{ccc}E'&{\xrightarrow[{}]{f'}}&E\\{\Bigg \downarrow }h'&&{\Bigg \downarrow }h\\C&{\xrightarrow[{}]{f}}&D\end{array}}}$

and a well-defined mapping is obtained by means of the assignment , which is designated by. In this way the so-called sub-object functor is obtained . Under the above-mentioned conditions, it is now true that there is a sub-object classifier if and only if the sub-object functor can be represented , more precisely if there is an object in and in natural isomorphisms ${\displaystyle h\mapsto h'}$ ${\displaystyle \mathrm {Sub} _{\mathcal {C}}(D)\rightarrow \mathrm {Sub} _{\mathcal {C}}(C)}$ ${\displaystyle \mathrm {Sub} _{\mathcal {C}}(f)}$ ${\displaystyle \mathrm {Sub} _{\mathcal {C}}:{\mathcal {C}}^{op}\rightarrow {\mathcal {Set}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle C}$

${\displaystyle \alpha _{C}:\mathrm {Sub} _{\mathcal {C}}(C)\cong \mathrm {Hom} _{\mathcal {C}}(C,\Omega )}$.

Individual evidence

1. ↑ Saunders Mac Lane , Ieke Moerdijk : Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , definition in chap. I.3
2. ↑ ^{a b} Saunders Mac Lane, Ieke Moerdijk: Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , chap. I.4
3. ↑ Peter T. Johnstone : Sketches of an Elephant, A Topos Theory Compendium, Volume 1 , Clarendon Press Oxford (2002), ISBN 978-0-19-853425-9 , Lemma A.1.6.6
4. ↑ Saunders Mac Lane, Ieke Moerdijk: Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , sentence 1 in chap. I.3