Sieve (category theory)

In the mathematical branch of category theory, a sieve denotes a set of morphisms with a fixed common goal and a certain legal ideal.

definition

Let there be a category and an object . A sieve open is a set of morphisms , where denotes the domain of definition , so that the following condition is fulfilled: ${\displaystyle {\mathcal {C}}}$ ${\displaystyle C}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle C}$ ${\displaystyle S}$ ${\displaystyle f:\mathrm {dom} (f)\rightarrow C}$ ${\displaystyle \mathrm {dom} (f)}$ ${\displaystyle f}$

Are and a morphism in so is . ${\displaystyle f\in S}$ ${\displaystyle g:\mathrm {dom} (g)\rightarrow \mathrm {dom} (f)}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle f\circ g\in S}$

Every composition of a sieve element with a further morphism from the right is therefore back in the sieve.

Simple examples and properties

• The empty set of morphisms is a sieve. ${\displaystyle S=\emptyset }$
• The set of all morphisms with a goal is one sieve, it is the maximum sieve on . ${\displaystyle C}$ ${\displaystyle C}$
• Averages and unions of seven are again sieves.
• If there is an arbitrary set of morphisms with aim , the average of all comprehensive sieves is the sieve produced by and it is ${\displaystyle R}$ ${\displaystyle C}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle (R)}$

${\displaystyle (R)=\{f\circ g\mid f\in R,g{\text{ ein mit }}f{\text{ komponierbarer Morphismus}}\}}$.

• Be a sieve on and a morphism so is ${\displaystyle S}$ ${\displaystyle C}$ ${\displaystyle f:D\rightarrow C}$

${\displaystyle f^{*}S:=\{h\mid f\circ h\in S\}}$

a sieve on which means on retracted sieve. ${\displaystyle D}$ ${\displaystyle f}$ ${\displaystyle D}$

• If a sieve is open and there are and morphisms, then applies . ${\displaystyle S}$ ${\displaystyle C}$ ${\displaystyle f:D\rightarrow C}$ ${\displaystyle g:E\rightarrow D}$ ${\displaystyle (f\circ g)^{*}(S)=g^{*}(f^{*}(S))}$

Sieves on topological spaces

It is a topological space . Then the category of open sets and inclusions is formed , which means that the objects in this category are the open ones and the only morphisms are the inclusions , more precisely the inclusion images . This can be used to identify target morphisms with open subsets . ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}(X)}$ ${\displaystyle U\subset X}$ ${\displaystyle V\subset U}$ ${\displaystyle \iota _{V,U}:V\rightarrow U}$ ${\displaystyle U}$ ${\displaystyle V\subset U}$

Then a sieve is on nothing more than a set of open subsets , so that every open quantity contained in a sieve set is also contained in the sieve. This clearly means: If an open amount fits through the sieve, then also every smaller one. ${\displaystyle U}$ ${\displaystyle V\subset U}$

If there is a system of open sets in , for example, an open covering of , the sieve generated by is obtained by adding all open subsets of the individual . Many constructions in sheaf theory over a topological space only use open coverages and their properties. The concept of the sieve was introduced in order to be able to generalize this to any categories. This is how you come to the concept of the Grothendieck topology . ${\displaystyle (V_{i})_{i\in I}}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle (V_{i})_{i\in I}}$ ${\displaystyle V_{i}}$

Sieves as sub-functors of the Hom functor

Each sieve on in a category defines a functor in the category of sets as follows : ${\displaystyle S}$ ${\displaystyle C}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle F_{S}:{\mathcal {C}}\rightarrow {\mathcal {Set}}}$

• For an object in be ${\displaystyle D}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle F_{S}(D):=S\cap \mathrm {Hom} (D,C)}$
• For a morphism in let be defined by . Apparently the diagram is ${\displaystyle g:D\rightarrow E}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle F_{S}(g):S(E)\rightarrow S(D)}$ ${\displaystyle (F_{S}(g))(h):=h\circ g}$

${\displaystyle {\begin{array}{ccc}S(E)&\subset &\mathrm {Hom} (E,C)\\\quad {\Bigg \downarrow }F_{S}(g)&&\quad \quad {\Bigg \downarrow }\mathrm {Hom} (g,C)\\S(D)&\subset &\mathrm {Hom} (D,C)\end{array}}}$

commutative such that is a sub-function of . ${\displaystyle F_{S}}$ ${\displaystyle \mathrm {Hom} (-,C):{\mathcal {C}}\rightarrow {\mathcal {Set}}}$

Conversely, for each Unterfunktor of a sieve. Therefore one usually identifies with and uses the sieve itself like a functor, namely like . ${\displaystyle \bigcup \{F(D)\mid D\in {\mathcal {C}}\}}$ ${\displaystyle F}$ ${\displaystyle \mathrm {Hom} (-,C)}$ ${\displaystyle S}$ ${\displaystyle F_{S}}$ ${\displaystyle F_{S}}$

In the textbook by H. Schubert given below , sieves are defined as sub-functors of Hom functors.

Individual evidence

1. ↑ Saunders Mac Lane , Ieke Moerdijk : Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , chap. I.4
2. ^ SI Gelfand , YI Manin : Methods of Homological Algebra , Springer-Verlag 1996, ISBN 978-3-662-03222-0 , chap. II, §4, definition 13
3. ↑ Saunders Mac Lane , Ieke Moerdijk : Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , chap. III.2: Grothendieck topologies
4. ^ H. Schubert: Categories II , Springer-Verlag (1970), ISBN 978-3-540-04866-4 , definition 20.1.2