Last Updated：2025/12/31
Sentence
Let
X
be
a
topological
space.
(Following
tradition,
I
will
switch
from
my
previous
convention
of
using
X
to
denote
an
object
of
a
topos.)
Write
Open(X)
for
its
poset
of
open
subsets.
A
presheaf
on
X
is
a
functor
F:
mathbf
Open(X)op→
mathbf
Set.
It
assigns
to
each
open
subset
U
a
set
F(U),
whose
elements
are
called
sections
over
U
(for
reasons
to
be
explained).
It
also
assigns
to
each
open
V⊆U
a
function
F(U)→F(V),
called
restriction
from
U
to
V
and
denoted
by
s→s|_V.
I
will
write
Psh(X)
for
the
category
of
presheaves
on
X.
Quizzes for review
Let X be a topological space. (Following tradition, I will switch from my previous convention of using X to denote an object of a topos.) Write Open(X) for its poset of open subsets. A presheaf on X is a functor F: mathbf Open(X)op→ mathbf Set. It assigns to each open subset U a set F(U), whose elements are called sections over U (for reasons to be explained). It also assigns to each open V⊆U a function F(U)→F(V), called restriction from U to V and denoted by s→s|_V. I will write Psh(X) for the category of presheaves on X. Examples 3.1 i. Let F(U) = {continuous functions U→ℝ}; restriction is restriction.
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