# Traces of nilpotent matrices

I needed the following well-known result in the course I’m teaching:

Lemma. Let be a field and a nilpotent matrix. Then .

The classical proof uses the flag of subspaces

to produce a basis for which is upper triangular. Here is a slick basis-independent commutative algebra proof that shows something better:

Lemma. Let be a commutative ring with nilradical , and let be a nilpotent matrix. Then the characteristic polynomial satisfies

Here we write for the polynomials in of degree smaller than whose coefficients lie in a given ideal .

Note that the formulation should ring a bell: in the previous post we saw that . When is a domain, this reduces to , and the lemma just says that .

This suggests that we shouldn’t work with but with its anadrome (or reciprocal) .

Proof of Lemma. We have to show that . Since is nilpotent, there exists with , so . Thus . Evaluating at shows that the constant coefficient is 1.

# Invertible polynomials

Here is a classic little lemma on polynomials:

Lemma. Let be a commutative ring with nilradical . Then .

That is, a polynomial is invertible if and only if is invertible and are nilpotent. This is the formulation you find in Atiyah–MacDonald, exercise 1.2(i). Note that when is a domain (or more generally a reduced ring), this just says that (via the constant polynomials).

Proof. First suppose with and . Multiplying by we may assume ; this does not change the nilpotence assumption on . Then is nilpotent since it is a sum of nilpotent elements in the commutative ring . Thus we get for some , and

shows that . Conversely, let , and suppose for some polynomial . Without loss of generality we may assume . If is a domain, then the leading term of is , so we must have and and are constant. In the general case, we conclude that for any ring homomorphism to a domain . In particular, taking for any prime ideal shows and . Taking the intersection over all prime ideals gives

# The 14 closure operations

I’ve been getting complaints that my lemmas have not been so lovely (or little) lately, so let’s do something a bit more down to earth. This is a story I learned from the book Counterexamples in topology by Steen and Seebach [SS].

A topological space comes equipped with various operations on its power set . For instance, there are the maps (interior), (closure), and (complement). These interact with each other in nontrivial ways; for instance .

Consider the monoid generated by the symbols (interior), (closure), and (complement), where two words in , , and are identified if they induce the same action on subsets of an arbitrary topological space .

Lemma. The monoid has 14 elements, and is the monoid given by generators and relations

Proof. The relations , , and are clear, and conjugating the first by shows that is already implied by these. Note also that and are monotone, and for all . A straightforward induction shows that if and are words in and , then . We conclude that

so (this is the well-known fact that is a regular open set). Conjugating by also gives the relation (saying that is a regular closed set).

Thus, in any reduced word in , no two consecutive letters agree because of the relations , , and . Moreover, we may assume all occurrences of are at the start of the word, using and . In particular, there is at most one in the word, and removing that if necessary gives a word containing only and . But reduced words in and have length at most 3, as the letters have to alternate and no string or can occur. We conclude that is covered by the 14 elements

To show that all 14 differ, one has to construct, for any in the list above, a set in some topological space such that . In fact we will construct a single set in some topological space where all 14 sets differ.

Call the sets for the noncomplementary sets obtained from , and their complements the complementary sets. By the arguments above, the noncomplementary sets always satisfy the following inclusions:
It suffices to show that the noncomplementary sets are pairwise distinct: this forces (otherwise ), so each noncomplementary set contains and therefore cannot agree with a complementary set.

For our counterexample, consider the 5 element poset given by

and let be the disjoint union of (the Alexandroff topology on , see the previous post) with a two-point indiscrete space . Recall that the open sets in are the upwards closed ones and the closed sets the downward closed ones. Let . Then the diagram of inclusions becomes We see that all 7 noncomplementary sets defined by are pairwise distinct.

Remark. Steen and Seebach [SS, Example 32(9)] give a different example where all 14 differ, namely a suitable subset . That is probably a more familiar type of example than the one I gave above.

On the other hand, my example is minimal: the diagram of inclusions above shows that for all inclusions to be strict, the space needs at least 6 points. We claim that 6 is not possible either:

Lemma. Let be a finite topological space, and any subset. Then and .

But in a 6-element counterexample , the diagram of inclusions shows that any point occurs as the difference for some . Since each of and is either open or closed, we see that the naive constructible topology on is discrete, so is by the remark from the previous post. So our lemma shows that cannot be a counterexample.

Proof of Lemma. We will show that ; the reverse implication was already shown, and the result for follows by replacing with its complement.

In the previous post, we saw that is the Alexandroff topology on some finite poset . If is any nonempty subset, it contains a maximal element , and since is a poset this means for all . (In a preorder, you would only get .)

The closure of a subset is the lower set , and the interior is the upper set . So we need to show that if and , then .

By definition of , we get , i.e. there exists with . Choose a maximal with this property; then we claim that is a maximal element in . Indeed, if , then so , meaning that there exists with . Then and , which by definition of means . Thus is maximal in , hence since and . From we conclude that , finishing the proof.

The lemma fails for non- spaces, as we saw in the example above. More succinctly, if is indiscrete and , then and . The problem is that is maximal, but and .

References.

[SS] L.A. Steen, J.A. Seebach, Counterexamples in Topology. Reprint of the second (1978) edition. Dover Publications, Inc., Mineola, NY, 1995.

# Finite topological spaces

One of my favourite bits of point set topology is messing around with easy topological spaces. What could be easier than finite topological spaces? The main result (below) is that the category of finite topological spaces is equivalent to the category of finite preorders.

Recall (e.g. from algebraic geometry) the following definition:

Definition. Let be a topological space. Then the specialisation preorder on (the underlying set of) is the relation if and only if .

Note that it is indeed a preorder: clearly , and if and , then , so , showing . We denote this preorder by .

Note that the relation is usually denoted in algebraic geometry, which is pronounced “ specialises to ”.

Definition. Given a preorder , the Alexandroff topology on is the topology whose opens are the cosieves, i.e. the upwards closed sets (meaning and implies ).

To see that this defines a topology, note that an arbitrary (possibly empty) union or intersection of cosieves is a cosieve. A subbase for the topology is given by the principal cosieves for any . We denote the set with its Alexandroff topology by .

Likewise, the closed sets in are the sieves (or downwards closed sets); for instance the principal sieves . The closure of is the sieve generated by ; for instance the closure of a singleton is the principal sieve .

Theorem. Let be the functor , and the functor .

1. Let be a preorder, a topological space, and a function. Then is a monotone function if and only if is a continuous function .
2. The functors and are adjoint: .
3. The composition is equal (not just isomorphic!) to the identity functor.
4. The restriction of to the category of finite topological spaces is equal to the identity functor.
5. If is a topological space, then is if and only if is a poset.
6. If is a preorder, then is a poset if and only if is .
7. The functors and give rise to adjoint equivalences

Proof. (1) Suppose is monotone, let be a closed subset, and let . Suppose and . Since is monotone and is closed, we get , i.e. . We conclude that , so is downward closed, hence closed in .

Conversely, suppose is continuous, and suppose in . Then , so by continuity we get , so .

(2) This is a restatement of (1): the map

is a bijection.

(3) Since , we conclude that if and only if , so the specialisation preorder on is the original preorder on .

(4) In general, the counit is a continuous map on the same underlying space, so is finer than . Conversely, suppose is closed, i.e. is a sieve for the specialisation preorder on . This means that if , then implies ; in other words . If and therefore is finite, there are finitely many such , so is the finite union

of closed subsets of . Thus any closed subset of is closed in , so the topologies agree.

(5) The relations and mean and . This is equivalent to the statement that a closed subset contains if and only if it contains . The result follows since a poset is a preorder where the first statement only happens if , and a space is a space where the second statement only happens if .

(6) Follows from (5) applied to since by (3).

(7) The equivalence follows from (3) and (4), and the equivalence then follows from (5) and (6).

Example. The Alexandroff topology on the poset is the Sierpiński space with topology . As explained in this post, continuous maps from a topological space to are in bijection with open subsets , where is sent to (and to the indicator function ).

Example. Let be a set with two elements. There are 4 possible topologies on , sitting in the following diagram (where vertical arrows indicate inclusion bottom to top):

These correspond to 4 possible preorder relations , sitting in the following diagram (where vertical arrows indicate inclusion top to bottom):

We see that finer topologies (more opens) have stronger relations (fewer inequalities).

Example. The statement in (4) is false for infinite topological spaces. For instance, if is the Zariski topology on a curve, then any set of closed points is downwards closed, but it is only closed if it’s finite. Or if is a Hausdorff space, then the specialisation preorder is just the equality relation , whose Alexandroff topology is the discrete topology.

I find the examples useful for remembering which way the adjunction goes: topological spaces generally have fewer opens than Alexandroff topologies on posets, so the continuous map should go .

Remark. On any topological space , we can define the naive constructible topology as the topology with a base given by locally closed sets for open and closed. In the Alexandroff topology, a base for this topology is given by the locally closed sets : indeed these sets are clearly naive constructible, and any set of the form for upward closed and downward closed has the property .

Thus, if is the Alexandroff topology on a preorder, we see that the naive constructible topology is discrete if and only if the preorder is a poset, i.e. if and only if is .

# Lawvere–Tierney topologies (topologies 6/6)

In the post of two weeks ago, we showed how Grothendieck topologies form a neat framework for the categorical aspects of the more familiar (at least to algebraic geometers) Grothendieck pretopologies. In this final post of the series, we take this one step further, to the notion of a Lawevere–Tierney topology on an arbitrary elementary topos .

Definition. An elementary topos is a category that has finite limits, is Cartesian closed, and has a subobject classifier (see last week’s post).

The only example we’re interested in today is the category of presheafs (of sets) on a small category , which we saw last week has subobject classifiers. But in fact, any Grothendieck topos (sheaves of sets on a (small) site) is an example of an elementary topos, so one could even introduce further topologies on those.

Example. The category of finite sets is an elementary topos, but not a Grothendieck topos since it is neither complete nor cocomplete.

Before giving the definition, we need to define one more structure on : the meet. Recall that the intersection (or meet) of two monomorphisms is the fibre product

The intersection of and is the monomorphism given by , which is classified by a map . Since is the universal monomorphism, we see that is the universal intersection of two subobjects, i.e. if and are classified by and respectively, then is classified by the composition

(If we denote this simply by , then is .)

Definition. Let be an elementary topos with subobject classifier . Then a Lawvere–Tierney topology on is a morphism such that the following diagrams commute:

We saw two weeks ago that a Grothendieck topology is a certain subpresheaf , and last week that is a subobject classifier on . Thus a subpresheaf is classified by a morphism , which we saw last week is given by .

Lemma. The subpresheaf is a Grothendieck topology on if and only if is a Lawvere–Tierney topology on . In particular, Grothendieck topologies on are in bijective correspondence with Lawvere–Tierney topologies on .

Thus Lawvere–Tierney topologies are an internalisation of the notion of Grothendieck topology to an arbitrary elementary topos .

Proof of Lemma. By definition of the morphism , we have a pullback square

The first commutative diagram in the definition above means that the top arrow has a section such that the composition is , i.e. as subobjects of . Since is the map taking to the maximal sieve for any , this means exactly that for all , which is condition 1 of a Grothendieck topology. For the second, consider the pullback

The condition means that as subobjects of . We already saw that for a Grothendieck or Lawvere–Tierney topology, so pulling back along gives . Thus the second diagram in the definition of a Lawvere–Tierney topology commutes if and only if , i.e. if with , then . But is given by , so this is exactly axiom 3 of a Grothendieck topology.

For the third diagram, we first claim that is monotone for all if and only if satisfies axiom 2 of a Grothendieck topology. Indeed, if is monotone and satisfy and , then the inclusion shows , so by axiom 3. Conversely, if satisfies axiom 2 and satisfy , then for any we have , so , i.e. .

The third diagram in the definition above says that the map given by is a morphism of meet semilattices. This implies in particular that is monotone, as if and only if , so the third diagram above implies axiom 2 of a Grothendieck topology.

Conversely, if is a Grothendieck topology, then axiom 2 implies that is monotone. In particular, for any , since . For the reverse implication, if satisfies , then and , so the remark of two weeks ago shows that , i.e. . We see that , showing that is a morphism of meet semilattices.

# Subobject classifiers on presheaf categories (topologies 5/6)

In the first post of this series, we saw how subobjects of representable presheaves correspond to sieves on . Last week, we saw how sieves give a convenient language for defining Grothendieck topologies on a small category. In next week’s (hopefully) final instalment of this series, we will generalise this even further to Lawvere–Tierney topologies on an arbitrary topos. Today’s post defines the last object we need to do this, which we will show generalises the presheaf from last week:

Definition. Let be a (possibly large) category with a terminal object . Then a subobject classifier on is a monomorphism in such that for every monomorphism in , there exists a unique arrow such that there is a pullback diagram

That is, is the “universal” monomorphism in , i.e. the pair represents the (possibly large) presheaf , where denotes isomorphism in the slice category . It is an easy exercise to show that any representative of this presheaf actually has the form described above, i.e. is a terminal object (apply the uniqueness property above to the identity monomorphism , and use the pullback square

coming from the hypothesis that is a monomorphism).

Example. If , then the two-point set with its natural inclusion given by is a subobject classifier: the monomorphism corresponds to the indicator function that is on and on its complement. (In other situations I would denote this by , but that notation was already used in this series to denote the representable presheaf .)

It’s even more natural to take to be the power set of . As in the first post of this series, we think of representing “true” and representing “false”. The generalisation of the power set of to presheaf categories is the presheaf of subpresheaves of defined last week:

Lemma. Let be a small category. Then the presheaf together with the map taking the unique section to the maximal sieve for any is a subobject classifier in .

Proof. Note that the prescribed map is a morphism of presheaves, since the inverse image of the maximal sieve under any morphism in is the maximal sieve . Again using the notation from last week, if is any monomorphism of presheaves, we get a morphism of presheaves defined on by

If is a morphism in , then for any we have

showing that , so is indeed a natural transformation. We already noted last week that for if and only if , so is the pullback

Conversely, if is any morphism with this property and , then if and only if , which together with naturality of gives

so .

We will discuss some other properties of subobject classifiers in future posts.

# Grothendieck topologies (topologies 4/6)

This post is the first goal in a series on sieves (subobjects of representable presheaves); I will give another generalisation in the next two posts. In the first post of the series, I defined sieves and gave basic examples, and last week I showed how the sheaf condition on a site can be stated in terms of sieves:

Corollary. Let be a (small) site. For a set of morphisms with the same target, write for the presheaf image of . Then a presheaf is a sheaf if and only if for every covering in , the inclusion induces an isomorphism

Thus, if is a site (a small category with a Grothendieck pretopology), we should be able to obtain the category of sheaves purely in terms of sieves. This is the notion of a Grothendieck topology that we describe at the end of this post.

Before giving the definition, note that any morphism in gives a pullback taking to its inverse image under (I avoid the word ‘pullback’ here to make sure this is truly a subpresheaf and not a presheaf with a monomorphism to defined uniquely up to unique isomorphism). Thus, is itself a presheaf (it takes values in since is small).

Also note the following method for producing sieves: if is a presheaf, a subpresheaf, and a section over some , we get a sieve by

By the Yoneda lemma, this is just the inverse image of along the morphism classifying . Note that is the maximal sieve if and only if .

Definition. Let be a small category. Then a Grothendieck topology on consists of a subpresheaf such that

1. For all , the maximal sieve is in .
2. If and with , then .
3. If is a sieve such that , then (equivalently, then is the maximal sieve ).

The sieves are called covering sieves. Since is a presheaf, we see that for any and any covering sieve , the pullback is covering. Condition 2 says that any sieve containing a covering sieve is covering. In the presence of condition 1, conditions 2 and 3 together are equivalent to the local character found in SGA IV, Exp. II, Def. 1.1:

• If with , such that for every morphism the inverse image of along is in , then .

Indeed, applying this criterion when immedately shows if , since the inverse image of along is the maximal sieve . Thus the local character implies criterion 2. The local character says that if contains a covering sieve , then is covering. Assuming criterion 2, the sieve contains a covering sieve if and only if is itself covering, so the local character is equivalent to criterion 3.

Remark. One property that follows from the axioms is that is closed under binary intersection, i.e. if then . Indeed, if for some , then

so . Axioms 2 and 3 give .

Example. Let be a pretopology on the (small) category ; see Tag 00VH for a list of axioms. For each , define the subset as those that contain a sieve of the form for some covering in . (See the corollary at the top for the definition of .) Concretely, this means that there exists a covering such that for all , i.e. is covered by morphisms that are in the given sieve .

Lemma. The association is a topology. It is the coarsest topology on for which each for is a covering sieve.

Proof. We will use the criteria of Tag 00VH. If , then there exists with . If is any morphism in , then by criterion 3 of Tag 00VH. But , because a morphism factors through if and only if factors through . Thus, , so , and is a subpresheaf of .

Condition 1 follows immediately from criterion 1 in Tag 00VH, and condition 2 is satisfied by definition. For condition 3, suppose satisfies . Then there exists with . This means that for all , i.e. for all . Thus, for each there exists in such that , i.e. for all and all . Thus, if denotes , then we get . But is a covering by criterion 2 of Tag 00VH, so .

If is any other Grothendieck topology for which each for is covering, then contains by criterion 2.

To state the obvious (hopefully), the notion of sheaf can therefore be defined on a Grothendieck topology in a way that coincides with the usual notion for a Grothendieck pretopology:

Definition. Let be a small category, and let be a Grothendieck topology. Then a presheaf is a sheaf if for any and any , the map induces an isomorphism

Thus, a Grothendieck topology is an internal characterisation (inside ) of which morphisms one needs to localise to get . In the last two posts, we will generalise this even further to a Lawvere–Tierney topology on an arbitrary topos.

# Covering sieves and the sheaf condition (topologies 3/6)

In the first post of this series, I explained how subobjects of the constant presheaf (resp. constant sheaf) on a small category (resp. small site) with terminal object correspond to left closed (resp. local) properties on . In this post, I will explain the main examples that intervene in setting up topoi, and show how to define the sheaf condition using sieves (instead of coverings).

For simplicity, assume is a small category with fibre products.

Definition. Given a set of morphisms with the same target , define the sieve generated by as the sieve on of those morphisms that factor through some .

It is in a sense the right ideal in generated by the . What does this look like as a subobject of ?

Example. If has one element, i.e. , then is the image of the morphism of representable presheaves . In the case where is already a monomorphism (this is always the case when is a poset, such as for some topological space ), then is itself injective (this is the definition of a monomorphism!), so is just .

In general, is the image of the map

induced by the maps . Indeed, an element of is a morphism , and it comes from some if and only if factors through .

This shows that, in fact, every sieve is of this form for some set : take as index set (the objects of) the slice category , which as in the previous post gives a surjection . This corresponds to generating an ideal by all its elements.

But we can also characterise without using the word ‘image’ (which somehow computes its first syzygy):

Lemma. Let be a set of morphisms with common target, and the sieve generated by . Then is the coequaliser of the diagram

where the maps are induced by the two projections .

We will give two proofs, one using the description of coequalisers of sets, and the other using that presheaves are colimits of representable presheaves, as discussed in the previous post.

Proof 1. The diagram

is a pullback, by the universal property of fibre products and since fibre products with a fixed set/presheaf of sets commute with coproducts. Then the same goes for the square

since is a monomorphism. But is an epimorphism (objectwise surjection) by definition, so this square is a pushout as well (in , epimorphisms are regular).

Proof 2. By the previous post, the presheaf is the colimit over of (see post for precise statement). Let be the diagram of the two projections, and let be the category of elements of , as in this post. There is a natural functor taking to and to , taking the morphisms in to the projections . We claim that is cofinal, hence the colimit can be computed over instead (see Tag 04E7).

To verify this, we use the criteria of Tag 04E6. If , then by definition the composition is given by a morphism that is contained in . Since is generated by the , this factors through some over , giving a map .

If and are two such maps, they factor uniquely through . The general result for and for (either of the form or of the form ) follows since elements of the form always map to the elements and , showing that the category is weakly connected.

Corollary. Let as above, and let be a presheaf on . Then

Proof. By the lemma above, we compute

so the result follows from the Yoneda lemma.

Corollary. Let be a (small) site. Then a presheaf is a sheaf if and only if for every object and every covering in the site, the inclusion induces an isomorphism

Proof. Immediate from the previous corollary.

Thus, the category of sheaves on can be recovered from if we know at which subobjects we should localise (make the inclusion invertible). Next week, we will use this to give a definition of a Grothendieck topology, abstracting and generalising the notion of a site (i.e. Grothendieck pretopology).

# Presheaves are colimits of representables (topologies 2/6)

Last week, I started a series on sieves and Grothendieck topoi. This is a short intermezzo on a well-known lemma from category theory that I will need for next week’s instalment.

Let be a small category. Recall that a presheaf on is a functor . Examples include the representable presheaves for , given by . The Yoneda lemma says that for any presheaf and any , the map

is an isomorphism. Applying this to shows that the Yoneda embedding

is fully faithful.

Given a functor , write for the comma category whose objects are pairs where and is a morphism (natural transformation) in . A morphism is a morphism such that the triangle

(1)

of natural transformations commutes, where denotes postcomposition by . Note that is again a small category, and there is a forgetful functor taking to .

By the Yoneda lemma, the category is isomorphic (not just equivalent!) to the category of pairs with and with morphisms given by morphisms in such that . It’s convenient to keep both points of view.

Lemma. Let be a small category, and let be a functor.

1. The object is naturally a cocone under via the morphisms .
2. This cocone makes the colimit of the diagram of representable functors.

In particular, any presheaf on a small category is a colimit of representable presheaves.

Proof. A cocone under is a presheaf with a natural transformation to the constant diagram with value . This means every is taken to a natural transformation , such that for any morphism , the square

commutes. By the Yoneda lemma, such a datum corresponds to an association of elements for all such that for every with , we have .

(1) To make a cocone under , simply take .

(2) Given any other cocone under , define the natural transformation by

Naturality follows since implies . It is clear that is the unique natural transformation of cocones under , showing that is the colimit.

One can also easily rewrite this argument in terms of natural transformations . For instance, the universal cocone is the natural transformation of functors given on by . Naturality of follows at once from (1). But checking that this thing is universal is a bit more tedious in this language.

Example. A standard example where this point of view is useful is simplicial sets. Let be the category of finite nonempty totally ordered sets, with (weakly) monotone increasing functions as morphisms. A simplicial set is a functor , and we often think of them as combinatorial models for topological spaces. The representable ones are the standard -simplices , where is the totally ordered set for .

If is a simplicial set, its value at is called the -simplices of . By the Yoneda lemma, this is . Then the story above is saying that a simplicial set is the colimit over all its -simplices for . This is extremely useful, as many arguments proceed by attaching simplices one at a time.

# Subterminal presheaves and sheaves (topologies 1/6)

Grothendieck pretopologies feature prominently in algebraic geometry, but the more beautiful concept of Grothendieck topologies is rarely touched upon. In a series of short posts, I aim to introduce some of these ideas, show how key concepts like the sheaf condition get very nice categorical descriptions in this language, and give examples of why topoi have much better formal properties than sites.

Let be a small category, and write for the functor category .

Definition. A sieve on an object is a subpresheaf of the representable presheaf .

Concretely, this means that each is a set of morphisms with the property that if is any morphism, then is in . Thus, this is like a “right ideal in “. Since is small, we see that sieves on form a set, which we will denote .

Lemma. Let and be small categories, an object, and a functor. Then there is a pullback map

defined by

If and is the forgetful functor, then gives a bijection

Proof. If is a sieve, then so is since and implies , so . For the second statement, given a sieve on , define the sieve on by

Then is a sieve on , and is the unique sieve on such that .

Beware that the notation could also mean the presheaf pullback , but we won’t use it as such.

Remark. In particular, it suffices to study the case where has a terminal object, which we will denote by (in analogy with the small Zariski and étale sites of a scheme , which have as a terminal object). We are thus interested in studying the subobjects of the terminal presheaf . We will do so both in the case of presheaves and in the case of sheaves. Note that is a sheaf: for any set (empty or not), the product is a singleton, so the diagrams

are vacuously equalisers whenever is a covering (or any collection of morphisms).

Definition. A property on a set is a function to the power set of a point . The property holds for if , and fails if .

Given a property on the objects of a small category , we say that is left closed if for any morphism , the implication holds. (This terminology is my own. Below, we confusingly prove that these are equivalent to what we described earlier as “right ideals”. This change of orientation arises from the fact that diagrams are drawn in the opposite direction compared to composition of morphisms.)

If is a site (a small category together with a Grothendieck pretopology), we say that is local if it is left closed, and for any covering in , if holds for all , then holds.

Lemma. Let be a small category with a terminal object .

1. Giving a subpresheaf of is equivalent to giving a left closed property on the objects of .
2. If is a site, then giving a subsheaf of the presheaf is equivalent to a giving a local property .

A homotopy theorist might say that a local property is a -truncated sheaf [of spaces] on .

Proof. 1. The terminal presheaf takes on values at every , thus any subpresheaf takes on the values and , hence is a property on the objects of . The presheaf condition means that for every morphism , there is a map , which is exactly the implication since there are no maps .

Alternatively, one notes immediately from the definition that a sieve on an object is the same thing as a subcategory of which is left closed.

2. Being a subpresheaf translates to a left closed property by 1. Then is a sheaf if and only if, for every covering in , the diagram

is an equaliser. If one is empty, then so is since is left closed, so the diagram is always an equaliser.

Thus, in the sheaf condition, we may assume for all , i.e. holds for all . Since is left closed, this implies that for all , so the two arrows agree on , and the diagram is an equaliser if and only if . Running over all coverings in , this is exactly the condition that is local.