Variable sets over an algebra of lifetimes:
a contribution of lattice theory to the study of computational topology
A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally persistence, is a central tool in topological data analysis, which examines the structure of data through topology. The basic techniques have been extended in several different directions, encoding topological features by so called barcodes or equivalently persistence diagrams. The set of points of all such diagrams determines a complete Heyting algebra that can explain aspects of the relations between persistent bars through the algebraic properties of its underlying lattice structure. In this paper, we investigate the topos of sheaves over such algebra, as well as discuss its construction and potential for a generalised simplicial homology over it. In particular, we are interested in establishing a topos theoretic unifying theory for the various flavours of persistent homology that have emerged so far, providing a global perspective over the algebraic foundations of applied and computational topology.
Keywords: Lattice theory, computational algebraic topology, topoi, sheaves over locales, persistent homology, persistence diagrams, semi-simplicial homology, topological data analysis.
AMS Mathematics Subject Classification: 03G30, 06D22, 18B25
Persistent homology is currently an area of research in computational and applied topology. The fundamental idea is to geometerize homology using multi-scale representations of spaces. One of the most prominent applications is topological data analysis, studying the topology of point clouds as a route for approximating topological features of an underlying geometric object generating the samples. Since the identification of persistent homology as the homology of graded -modules in , an algebraic approach has yielded immense benefits both in the innovation of new methods for topological data analysis and for algorithmic development. We identify the development of zigzag persistent homology , as well as progress made on multidimensional persistence  as coming from fundamentally algebraic considerations. However, the approach of using more complicated rings, as in , to model more general notions of persistence has raised a number of obstacles. In this paper, we adopt a different approach. We establish a foundation theory describing a general unifying framework for persistence with the construction of the appropriate topos of variable sets over an algebra of lifetime intervals . It is based on a theory of variable sets constructed over a lattice of time-like intervals of real numbers. This will permit us to compute homology over a category with similar structure to the category of sets, that is parametrised by the lifetime intervals leaving in the algebra . The intuition is that we can develop a set theory where all the sets have encoded a multiplicity of lifetimes determined by . In this setting, the topological features of a shape have lifetimes themselves. The goal of this approach is to place all flavours of persistence into a common framework, where some parameter of the framework determines the shape of the theory and the category in which analyses live. For approachable introductions to the field and its applications, we recommend ,  and , and for an accessible introduction to algebraic topology .
A sheaf of sets can be seen as a functor that is able to glue compatible local information providing us with a global perspective. A category of sheaves of sets is thus a collection of such functors and natural transformations between them. On the other hand, the topos we are interested in is the category of sheaves over . The category of sets, a base of most of mathematical and mathematic flavoured constructions, can be generalised by such topos: it is the topos of sheaves of sets on the one point space, The usefulness of sheaves for probabilistic reasoning in quantum mechanics was recognised by Abramsky and Brandenburger in , where they were able to generalise the no-go theorem by Kochen-Specker using a sheaf-based representation modelling contextually in quantum mechanics by sheaf-theoretic obstructions. Based on this, Döring and Isham establish a topos-theoretical foundation for quantum physics in  that is also described formally in the recent book of Flori on topos quantum theory in . The inspiration for our approach is to some extent rooted in the presentation of time sheaves in the exposition by Barr and Wells , where sheaves of sets are described as sets with a particular shape given by a temporally varying structure. The shape corresponds to the shape of the underlying site, which also describes the shape of the available truth values for the corresponding logic. Classical logic and set theory correspond to having two discrete truth values, while fuzzy logic corresponds to a continuum of truth values encoding reliability of a statement (e.g.: fuzzy sets are sheaves over ). Under this perspective, the persistent approach would encode truth as valid over some regions of a persistence parameter, but not other. In , the authors give examples of time-like structures modelled by sheaf theory: over the total order (sets have elements that arise and stay); and over intervals in (sets have elements that are born and die). The idea of applying sheaves to encode the shape of persistent homology is not itself new: it has been approached independently by McPherson and Patel , and by Ghrist and Curry in  and . Though, this research provides us with an approach that can encode the various flavours of persistent homology through the internal logic of the persistence. We believe that topos-theoretic perspective can provide such a unifying theory.
The fundamental observation is that we have seen numerous cases lately where the shape of a persistence theory matters; there has been the classical persistent homology as defined in , multi-dimensional persistence as defined in  and zig-zag persistence as defined in . In all of these cases, there is a sense of shape to the theory, embodied by a choice of algebra and module category that reflects the kinds of information we can extract from the method. The similarities in definitions and in algorithms suggest to us that all three should be instances of a unifying theory; and indeed, one suggests itself directly from the definitions: in all three cases, we study homology for graded modules over graded rings (see [35, 10, 9, 33] for details). However, this similarity steps in on the algebraic plane; we are interested in a unifying theory that connects the underlying topological cases as well. In , M. Vejdemo-Johansson reviews in more detail the algebraic foundations of persistent homology, and presents the idea of a topos-based approach as a potential unifying language for these various approaches. In this paper we will show how to encode the lifetimes of topological features with an Heyting algebra determined in the space of all possible points in a persistence diagram. Then, we generalise the underlying set theory by the construction of a topos of sheaves over providing the basis for our unifying theory. Such a topos can be seen as a category of sets with lifetime where things exist at some point and after a while might cease to exist. Later, we discuss the computation of simplicial and semi-simplicial homology over such category of sets with lifetimes. In such a setting, the vertices of the simplexes have themselves lifetimes encoded in the underlying algebra . Hence, this seems to be a more appropriate universe to deal with problems in unified theory of persistence.
A lattice is a poset for which all pairs of elements have infimum and supremum. Whenever every subset of a lattice has a supremum and a infimum, is named a complete lattice. Every total order is a lattice. Though, not all of them are complete: with the usual order does not include the supremum of all its elements. A lattice can be seen as an algebraic structure with two associative, commutative and idempotent operations and satisfying the absorption property, i.e., for all , . Moreover, if and only if if and only if , for all , providing the equivalence between the algebraic structure of a lattice and its ordered structure. Given a lattice , an ideal of is a nonempty subset of closed to such that, for all and , implies . A filter is defined dually. The ideal [filter] generated by a singleton , with , is called principal ideal [filter] and denoted by . An ideal [filter] of distinct from is called proper. A proper ideal is maximal in if there is no other ideal in containing . A proper ideal is prime if, for all , implies or . A prime filter is defined dually. In fact, is a prime ideal if and only if is a prime filter. An element of is prime if is a principal prime ideal. An element of a lattice is join-irreducible if for all such that , we have or . In a distributive lattice , a nonzero element is join-irreducible if and only if is a prime ideal. Dually, a nonzero element is meet-irreducible if and only if is a prime filter (cf. ).
A lattice is distributive if, for all , it satisfies . A Boolean algebra is a distributive lattice with a unary operation and nullary operations and such that for all elements , and ; and . A bounded lattice is a Heyting algebra if, for all there is a greatest element such that . This element is the relative pseudo-complement of with respect to denoted by . Please notice that we will distinguish the notation of this operation from the notation of logic implication, denoting the latter by a long right arrow . A subalgebra of an Heyting algebra is thus closed to the usual lattice operations and , and to . A homomorphism between Heyting algebras must preserve both lattice operations as well as the implication operation and both top and bottom elements. All the finite nonempty total orders (that are bounded and complete) constitute Heyting algebras, where equals whenever , and otherwise. Every Boolean algebra is a Heyting algebra, with given by . The lattice of open sets of a topological space forms a Heyting algebras under the operations of union , empty set , intersection , whole space , and the implication operation . A complete Heyting algebra is a Heyting algebra which constitutes a complete lattice. It can also be characterised as any complete lattice satisfying the infinite distributive law, i.e., for all and any family of elements of , , with the implication operation given by , for all .
Given a complete Heyting algebra and a contravariant functor over the category of sets, a compatible family in is a family of elements in such that for each pair and of the restrictions of and agree on the overlaps, i.e., . Moreover, is a sheaf if, given in and a compatible family in , there exists a unique element such that for each index , . Equivalently is a sheaf if it satisfies the following two conditions:
Given , if is a family of elements in such that , and if are such that for each , then (we then call a separated presheaf);
Given , if is a family of elements in such that , every compatible family can be ”glued” into a section such that for each .
The first condition is usually called Locality while the second is called Gluing. By the first condition, is unique. Thus, both conditions together state that compatible sections can be uniquely glued together. For any objects and in a category with all binary products with , an exponential object is an object of equipped with an evaluation map such that for any object and map there exists a unique map such that equals the map . Whenever existing, it is unique up to unique isomorphism. The subobject classifier, , is an object of and a monomorphism (where is the terminal object) such that for every monomorphism in there is a unique morphism determining the correspondent pullback diagram. A category is Cartesian closed if it has a terminal object and, for each pair of objects and in , the product and the exponential object exist. Any Heyting algebra seen as a thin small (poset) category is Cartesian closed: for all , the product of and is and the exponential is . A topos is a Cartesian closed category with all finite limits and a subobject classifier. The category is a topos with subobject classifier . Roughly speaking, any topos behaves as a category of sheaves of sets on a topological space. If is a topological space, the category of sheaves over , , is a topos with . In general, the category of sheaves over a complete Heyting algebra is a topos (cf. ). A good review on Heyting algebras, sheaf theory and, in particular, on topos theory can be found in ,  and .
2. Motivation on Persistent Homology
As motivation for this research, we describe some aspects of computational topology with a focus on the computation of persistent homology, as well as review some of the variants. Persistent homology permits us to recover topological information by applying geometric tools followed by methods from algebraic topology to obtain a topological descriptor. In topological data analysis we often view data as a finite metric space, build complexes of points (most often Čech or Vietoris-Rips), and analyse the topology of these objects to infer the topology of that data. Recall that, due to the nerve theorem, the Čech complex associated with any covering of the space with balls is homotopy equivalent to the original space. The construction of these complexes requires a choice of parameter (such as the radius of the balls for the Čech complex). Persistent homology lets the parameter value vary while tracking the births and deaths of topological features. The output, in the standard case from  is a set of intervals on the real line that can also be encoded as points in a persistence diagram. This measures the significance of a topological feature. Usually additional restrictions are imposed to ensure that the homology changes at only finitely many values. Many of these restrictions can be relaxed, defining persistence diagrams in a wide variety of situations as in .
A -dimensional homology class is an equivalence of -cycles, i.e., a collection of mutually homologous points (in dimension ), closed curves (in dimension ) or closed surfaces (in dimension ) in . The -th homology group of the space is the vector space of all -dimensional homology classes with rank , the -th Betti number of . If is a subspace of , the -th relative homology group of the pair of spaces is the vector space of all -dimensional relative homology classes with rank , the -th relative Betti number of . The essential classes are the ones that represent the homology of while all others are called inessential classes. As homology classes do not come with a notion of size, persistent homology takes a compact topological space along with a real-valued (height) function and returns the size, as measured by , of each homology class in .
Let be a space and a real function. We denote by . The object of study of persistent homology is a filtration of , i.e., a monotonically non-decreasing sequence
determined by the (height) function as follows: . As runs in , the sublevel sets include into one another and get bigger, eventually forming the space itself, while homology classes appear and disappear. Persistent homology is the homology of a filtration, tracking and quantifying the described evolution. To simplify the exposition, we assume that this is a discrete finite filtration of tame spaces. Taking the homology of each of the associated chain complexes, we obtain
We take homology over a field – therefore the resulting homology groups are vector spaces and the induced maps are linear maps. The standard persistent homology module describes how the absolute homology groups relate to each other as varies. Due to a version of Alexander duality , a similar description is possible for the absolute cohomology groups , the relative homology groups , and the relative cohomology groups as represented below:
To each homology class is assigned a lifetime encoded by an interval with endpoints in the real line. To each dimension, encoded as a non-negative integer k, corresponds a persistence barcode, by which we mean the finite collection of lifetimes of the homology classes appearing in its filtration. The integer specifies a dimension of a feature (zero-dimensional for a cluster, one-dimensional for a loop, etc.), and any bar in the barcode represents a feature which is born at the value of a parameter given by the left hand endpoint of the interval, and which dies at the value given by the right hand endpoint. A persistence diagram is a multi-set of points of representing a persistence barcode by corresponding a bar in the barcode with birth time and death time by the point .
The persistence diagram for absolute cohomology (as for relative homology and cohomology) is also a multi-set of integer ordered pairs. Moreover, homology and cohomology have identical barcodes, while persistent homology and relative homology barcodes carry out the same information, with a dimension shift for the finite intervals. Thus, provided we take the dimension shifts into account, all four barcodes carry exactly the same information (cf. ).
Given a real-valued function , we call extended persistence to the collection of pairs arising from a sequence of absolute and relative homology groups. The correspondent pairs in the extended persistence diagram keep track on the changes in the homology of the input function. As in , we consider persistent homology as a two-stage filtering process:
standard persistence: In the first stage, filter via the sub level sets of , where ;
extended persistence: In the second stage, we consider pairs of spaces , where is a super level set and .
Every class which is born at some point of the two-stage process will eventually die, being associated with a pair of critical values. These pairs fall into three types:
ordinary pairs : have birth and death during the first stage, being represented in the persistence diagram by a point with ;
relative pairs : have birth and death during the second stage, being represented in the persistence diagram by a point with ;
extended pairs : have birth in one stage and death in the other (their representation in the persistence diagram will coincide with one of the two cases above as seen in Fig. 1.
In Figure 1 it is represented a version of the classical case of a torus with a height function from , together with the correspondent barcode comprehending the ordinary classes given by the bars in a positive (upwards) direction; the relative classes given by the bars in the negative (downwards) direction; and the extended classes in which the bars go to infinity and then come back. Also in the same figure its represented (on the right) the traditional persistence diagram that tracks the topological information given by the barcodes. Notice that the ordinary classes are represented by points where , while relative classes are represented by points where .
It is possible to distinguish both the multiplicity of an element in a persistence diagram, or the indication that such element corresponds to an extended bar or not. Though, in the following sections we shall consider the space of all ordinary and relative pairs in a persistence diagram, ignoring their multiplicity and identifying these pairs with the extended pairs that have the same coordinates. We shall distinguish between relative and ordinary pairs, corresponding to bars with different orientation. Other possible directions of research point out to have the algebra of lifetimes, described in the next section, being build over aspects of total and pointwise existence of persistence. This provides new arguments for the choice of this model and is a research topic by itself to be developed in further steps.
In the variants described above, the methods are tied to the total order of the reals . Generalisations of persistence such as zigzag persistence  or multidimensional persistence  do consider other oreder underlying structures. However, these theories are much less developed. The purpose of this investigation is to explore the underlying lattice structure and introduce a unifying framework where these (and other) generalisations of persistence can find a common language.
3. The algebra of lifetimes
Our interest is to model the algebra of barcodes used in the methodology of persistent homology, considering non disjoint intervals, i.e., time indexed sets. In this, given a time indexed set , is the birth time while is the death time. With this in mind, the set of all intervals of real numbers, ordered by set inclusion, corresponds to the algebra of open sets of the topological space and constitutes a complete Heyting algebra for the operations of set theoretic intersection and union. However it also includes disjoint intervals which are not of interest in our model, in the sense that a lifetime should correspond to a closed interval of the real line, having a birth time and a death time. Fix and consider the total order in the positive real numbers no bigger than including zero, i.e., the complete lattice where and , for any . If we substitute set theoretic union by its cover (i.e., ), we get a complete lattice. It unfortunately does not constitute a Heyting algebra as distributively fails: to see this consider the intervals , and and observe that and that
Consider now the representation of barcodes in a persistence diagram. Let be the quarter plane of all the points in all possible persistence diagrams bounded by and . Let be a point in a persistence diagram and denote by the correspondent interval , where is the birth time and is the death time. Recall from Remark 2.1 that we are also considering points of the persistence diagram with coordinates such that . Consider the set of points of all possible persistence diagrams. Then, with the operations
constitutes a lattice ordered by
for all .
These operations give to the algebraic structure of a lattice, named the algebra of lifetimes. In fact considering the non negative real numbers in for with the usual partial order as the poset category where a morphism is just , is given by the product . We will not distinguish the notation of theses lattice operations neither the correspondent order from the notations for lattice operations and order in whenever the context is clear. Figure 2 shows a representation of those operations from in the extended barcode and in the persistence diagram.
Each element of is an ordered pair that can be seen as a generalised time interval with:
a length, given by the absolute value of that corresponds to the persistence of the measured topological feature;
an orientation: positive if and negative if .
The degenerate case where corresponds to a lifetime of length zero.
The partial order determined by the lattice operations shows us how the operations are indeed natural: the derived order structure corresponds to the inclusion of the correspondent bars for the upper triangle where death times are smaller then birth times: for all such that ,
This is not a total order: unrelated bars and are of such sort that and . Moreover, the smallest element is in the right lower corner of the diagram, correspondent to the point , while the biggest element is on the left upper corner, correspondent to the point . Furthermore, if one of the coordinates is equal, then the correspondent bars are always related. Hence, bars with equal death time or birth time must be included in one another.
The algebra of , together with the extended operations
determines a complete lattice satisfying the infinite distributive law given by the following identity:
for all and any family of elements of . Hence, is a complete Heyting algebra.
By construction, the lattice operations can naturally be generalised from pairs of elements to any set of elements. The infinite distributivity law follows directly from the definition of the lattice operations together with the fact that constitutes a completely distributive lattice. To see this just observe that:
The fact that is a Heyting algebra follows from being a complete and distributive lattice satisfying the infinite distributivity law (cf. ). ∎
Let us now examine the implication operation. The following result describes this operation for any of the possible cases, represented in Figure 3, both in the context of an extended barcode or a persistence diagram.
Let . Then, if and are (order) related,
The fact that the persistence diagram is a complete distributive lattice implies that the implication operation is defined for every as follows:
Case 1: When we get that and so that and can take any value. The biggest bar in these conditions has the smallest and the biggest , thus coinciding with the maximum element .
Case 2: On the other hand, if then and so that can only hold when and . The biggest interval in this conditions corresponds to and .
Case 3: When and , then and so that
Case 4: Similarly, whenever and , then and . Therefore,
Due to the completeness of the underlying lattice structure, the operations and are adjoints in two suitable monotone Galois connections. Particularly, the fact that is a Heyting algebra implies that, given , the mapping defined by is the lower adjoint of a Galois connection with respective adjoint defined by . This Galois connection defines a pair of dual topologies so that such topologies can be defined by means of binary relations (cf. ). Given , if and only if . But means that and . Thus
Then, for all ,
These are thus the conditions for candidates in the algebra to be the element .
Observe that the algebra of lifetimes has a largest element given by , for any . Moreover, is not a Boolean algebra: indeed, take such that , and (recall that and are the biggest values on the coordinates and coordinates). Then, for all , implies and , that is, and . Similarly, implies and , that is, and . Hence, has no complement in . In fact, the only complemented elements in are , , and .
The following paragraphs describe the order structure of through the study of its ideals and filters. Towards the end of this section, we shall also discuss aspects of the dual space for in the light of Stone duality. An ideal of the algebra of lifetimes is any downset that constitutes a subalgebra, i.e., any subset closed to the operation such that, for all and , implies . Notice that an ideal of closed to arbitrary joins must be determined by one unique element, i.e., it must be a principal ideal. Given an arbitrary element , the ideal it generates is the following set:
Dually, the principal filter generated by is
The ideal generated by two elements and of , , is the ideal generated by , i.e., Dually, the filter generated by two elements and of , , is the filter generated by , i.e., In general, for any family of elements of , we get that:
Take two arbitrary elements and of . Let and . Then and , so that . On the other hand, if then and , so that The dual result has a similar proof and the general case can be proven by induction on the number of generating elements, taking in account the completeness of the lattice. ∎
The join-irreducible elements of are all the elements or , with and . Dually, the meet-irreducible elements are all the elements and , with and .
Given an arbitrary element , the elements and are such that . Thus, if and are distinct from , then is not a join-irreducible element. Hence, the candidates for join-irreducible elements are the bars such that is the biggest first coordinate, i.e., ; or is the least second coordinate, i.e., . Let us see that this indeed is the case. Take such that for any and in . Then , i.e., and and thus or so that or . Similarly, implies , i.e., . On the other hand, or so that or . The proof regarding meet-irreducible elements is analogous. ∎
All prime principal ideals are of the form or , for some . Dually, all prime principal filters are of the form or , for some .
Let . It follows from the distributivity of that is irreducible if and only if is a prime ideal (cf. pp.63). Dually, is -irreducible element of if and only if is a prime filter. On the other hand, is a prime filter if and only if is a prime ideal of (cf. pp.25). Then, is -irreducible element of if and only if is a prime ideal of . Hence, all prime principal ideals are determined by an element of the form or , for some . ∎
The category of locales is determined by complete Heyting algebras (as objects) and morphisms between them preserving finite and arbitrary . A sober space is a topological space such that every irreducible closed subset of has a unique point whose closure is all of . A locale is spatial (or topological) if each element of can be expressed as a meet of prime elements.
The algebra of lifetimes, , is a spatial locale.
Any element of the algebra of lifetimes is determined by the prime ideals corresponding to its coordinates: considering and , where the prime principal ideals correspondent to and are and , respectively, represented in Figure 5. ∎
A lattice duality describes the categorical equivalence between the category of topological spaces that are sober with continuous functions, and the category of locales that are spatial with appropriate homomorphisms (cf. ). The dual space correspondent to the locale is made out of all its prime elements, i.e., all the elements in that determine principal prime ideals: and , for all . A basis of opens for the topology of that dual space is given by the vertical and horizontal filters and . Each filter, horizontal or vertical, must contain to give a sober space. The open sets are exactly the unions of these filters with different behaviours and thus we get a direct sum topology. This is exhibited in Figure 6.
The topological space described considers as points, birth and death time instances such as points in the diagonal, corresponding to an instance of time in some lifetime (open set) in . This is in concordance with the fact that a real in is a point in the locale , and that . That provides us with the concept of point of the locale corresponding to a prime element in as earlier described. The spaciality given by the proof of Proposition 3.8 is based on the recover of every element of the lattice as a lifetime corresponding to an interval with a birth time and a death time . Both of these instances are given by the corresponding open determined by both filters and .
The discussed duality can provide more efficient techniques to solve problems living in the algebra of lifetimes , by dealing with them in the dual space and transferring the solutions back to the lattice. Such techniques and their implementation are subject of future work.
4. From local to global
In the following section we will discuss the construction of sheaves over the algebra , and will have a further look at the correspondent Grothendieck topos that will determine a framework of sets with a lifetime. Recall that the barcode (or the persistence diagram) for a persistence module encodes the basis elements of the persistence module as pairs of a birth point and a death point of the basis element. This boils down to a persistence diagram or barcode being a multi-set of pairs of real numbers (or whatever the time set happens to be). Each element of the multi-set corresponds to a basis element of the module. One of the ways this is being used is to say that for a given time point , we can determine the local Betti number at that time by counting the number of points in the multi-set such that . This can be visualised either as counting points in a quadrant or as counting bars intersecting a vertical line. There is nothing that keeps us from doing this for longer spans of query time intervals – we can ask for points such that for some interval . This produces Betti numbers that persist for at least the time period . Now, the persistence Heyting algebra has as its elements intervals , ordered by inclusion and with a Heyting algebra structure built according to the constructions in this text. Any actual barcode can be considered as a sheaf over this Heyting algebra such that is the collection of basis elements in the persistence module that exist at least in the entire interval . In this setting, sheaves over encode (extended) barcodes or persistence diagrams. These sheaves can be considered as sets where each element is visible only over some time(-like) intervals.
The category of sheaves over , denoted by , is defined as follows:
Every object of this category is a sheaf , and every morphism between two functors is a natural transformation .
As these functors are contravariant, is an assignment to every object of a morphism in (usually called the component of at ) such that for any poset arrow in , the following diagram commutes in :
Consider the presheaf defined by the sections , for all , and the restriction map defined by for all such that . Then is a sheaf of sets over .
It is clear that, in general, so that the compatibility condition reduces to . Let us now show that is a separated presheaf. Given in and , the infinite distributivity law and the identity together imply that
We shall now see that constitutes a sheaf: given in and a compatible family such that
put and notice that
Hence, , as required. ∎
The category of sheaves is a topos, with subobject classifier determined by the set of all sieves on , that is,
with restriction maps given by , for all such that .
Let us notice that for , corresponds to all the topological features that live while a feature with lifetime is alive. Another presheaf that can be considered on is determined by , for all , and by the restriction map defined as for all such that . can be seen as the inclusion of in the larger , corresponding to the example of the time sheaf of states of knowledge described in . Recall that, in the poset category , is the terminal object. Now observe that the category also has terminal object: the constant functor which sends every element of to is a terminal object . It also has finite limits and exponentials due to the fact that the Yoneda embedding preserves all products and exponentials in , and the fact that limits and exponentials exist in any Heyting algebra (corresponding to the meet and arrow operations). While the terminal object of is a constant sheaf, all exponential objects in constitute sheaves themselves. And since limits commute with limits and every sheaf can be seen as an equaliser, the limits of sheaves are also sheaves (cf. ). Now, to show that is a topos we just need to show that as defined in is the subobject classifier. Observe that is the set of all (poset) arrows into a given . Consider the morphism between the terminal object and , i.e., a natural transformation , that takes the maximal sieve for each , sending the point of to the maximum element of . Now let be a given monomorphism and take the morphism