Let
- 1.
Functoriality. The assignment
defines a functorIn particular, for each
, the following condition is satisfied:- If
, then .
- If
- 2.
Oplax Associativity. We have a natural transformation
with components
for each
. - 3.
Left Unitality. The diagram
commutes, i.e. we have
for each
. - 4.
Oplax Right Unitality. The diagram
does not commute in general, i.e. we may have
in general, where
. However, when is nonempty, we have - 5.
Interaction With Unions I. The diagram
commutes, i.e. we have
for each
. - 6.
Interaction With Unions II. The diagram commute, i.e. we have
for each
and each . - 7.
Interaction With Intersections I. We have a natural transformation
with components
for each
. - 8.
Interaction With Intersections II. The diagrams commute, i.e. we have
for each
and each . - 9.
Interaction With Differences. The diagram
does not commute in general, i.e. we may have
in general, where
. - 10.
Interaction With Complements I. The diagram
does not commute in general, i.e. we may have
in general, where
. - 11.
Interaction With Complements II. The diagram
commutes, i.e. we have
for each
. - 12.
Interaction With Complements III. The diagram
commutes, i.e. we have
for each
. - 13.
Interaction With Symmetric Differences. The diagram
does not commute in general, i.e. we may have
in general, where
. - 14.
Interaction With Internal Homs I. The diagram
does not commute in general, i.e. we may have
in general, where
. - 15.
Interaction With Internal Homs II. The diagram
commutes, i.e. we have
for each
and each . - 16.
Interaction With Internal Homs III. The diagram
commutes, i.e. we have
for each
and each . - 17.
Interaction With Direct Images. Let
be a map of sets. The diagramcommutes, i.e. we have
for each
, where . - 18.
Interaction With Inverse Images. Let
be a map of sets. The diagramcommutes, i.e. we have
for each
, where . - 19.
Interaction With Direct Images With Compact Support. Let
be a map of sets. The diagramcommutes, i.e. we have
for each
, where . - 20.
Interaction With Unions of Families I. The diagram
commutes, i.e. we have
for each
. - 21.
Interaction With Unions of Families II. Let
be a set and consider the compositionsgiven by
for each
. We have the following inclusions:All other possible inclusions fail to hold in general.