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Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We construct the equivariant version of cd-structures, and we develop descent theory for topologies coming from equivariant cd-structures. In particular, we reprove several results of Cisinski–Déglies on étale descent, qfh-descent, and h-descent. Since the étale topos, qfh-topos, and h-topos do not come from usual cd-structures, such results cannot be produced by usual cd-structures. We also apply equivariant cd-structures to study several topologies on the category of noetherian fs log schemes. [ABSTRACT FROM AUTHOR]

Abstract We give simple characterizations of the category PAsm (A) of partitioned assemblies , and of the realizability topos RT (A) over a partial combinatory algebra A. This answers the question for an 'extensional characterization' of realizability toposes. [ABSTRACT FROM AUTHOR]

Abstract We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or "structural" set theories in the spirit of Lawvere's "Elementary Theory of the Category of Sets". We consider weak intuitionistic and predicative theories of pretoposes, and we also propose category-theoretic versions of stronger axioms such as unbounded separation, replacement, and collection. Finally, we compare all of these theories formally to traditional membership-based or "material" set theories, using a version of the classical construction based on internal well-founded relations. [ABSTRACT FROM AUTHOR]

Copyright of Revue Nerval is the property of Classiques Garnier and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let ${\cal E}$ be a topos, ${\rm{Dec}}\left({\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many 'gros' toposes in Algebraic Geometry, simplicial sets and other toposes of 'combinatorial' spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry. [ABSTRACT FROM AUTHOR]

In long lifespan software systems, specification documents can be outdated or even missing. Developing new software releases or checking whether some user requirements are still valid becomes challenging in this context. This challenge can be addressed by extracting high-level observable capabilities of a system by mining its source code and the available source-level documentation. This paper presents feature extraction and traceability (FEAT), an approach that automatically extracts topoi, which are summaries of the main capabilities of a program, given under the form of collections of code functions along with an index. FEAT acts in two steps: first, clustering: by mining the available source code, possibly augmented with code-level comments, hierarchical agglomerative clustering groups similar code functions. In addition, this process gathers an index for each function. Second, entry point selection: functions within a cluster are then ranked and presented to validation engineers as topoi candidates. We implemented FEAT on top of a general-purpose test management and optimization platform and performed an experimental study over 15 open-source software projects amounting to more than 1 M lines of codes proving that automatically discovering topoi is feasible and meaningful on realistic projects. [ABSTRACT FROM AUTHOR]

We give a characterisations of toposes which admit a generating set of objects which are internally cardinal finite (i.e. Kuratowski finite and decidable) in terms of “topological” conditions. The central result is that, constructively, a hyperconnected separated locally decidable topos admit a generating set of cardinal finite objects. The main theorem is then a generalisation obtained as an application of this result internally in the localic reflection of an arbitrary topos: a topos is generated by cardinal finite objects if and only if it is separated, locally decidable, and its localic reflection is zero dimensional. [ABSTRACT FROM AUTHOR]

We define a class of sites such that the associated topos is equivalent to the category of smooth sets (representations) of some locally prodiscrete monoids (to be defined). Examples of locally prodiscrete monoids include profinite groups and finite adele valued points of algebraic groups. This is a generalization of the fact that the topos associated with the étale site of a scheme is equivalent to the category of sets with continuous action by the étale fundamental group. We then define a subclass of sites such that the topos is equivalent to the category of discrete sets with a continuous action of a locally profinite group. [ABSTRACT FROM AUTHOR]

We show that a functor category whose domain is a colored category is a topos. The topos structure enables us to introduce cohomology of colored categories including quasi-schemoids. If the given colored category arises from an association scheme, then the cohomology coincides with the group cohomology of the factor scheme by the thin residue. Moreover, it is shown that the cohomology of a colored category relates to the standard representation of an association scheme via the Leray spectral sequence. [ABSTRACT FROM AUTHOR]

We study spans of cospans in a category C and explain how to horizontally and vertically compose these. When C is a topos and the legs of the spans are monic, these two forms of composition satisfy the interchange law. In this case there is a bicategory of objects, cospans, and 'monic-legged' spans of cospans in C. One motivation for this construction is an application to graph rewriting. [ABSTRACT FROM AUTHOR]

Topos quantum theory (TQT) is standardly portrayed as a kind of 'neo-realist' reformulation of quantum mechanics. In this article, I study the extent to which TQT can really be characterized as a realist formulation of the theory, and examine the question of whether the kind of realism that is provided by TQT satisfies the philosophical motivations that are usually associated with the search for a realist reformulation of quantum theory. Specifically, I show that the notion of the quantum state is problematic for those who view TQT as a realist reformulation of quantum theory. [ABSTRACT FROM AUTHOR]

We characterize Morita equivalence of theories in the sense of Johnstone in terms of a new syntactic notion of a common definitional extension developed by Barrett and Halvorson for cartesian, regular, coherent, geometric and first-order theories. This provides a purely syntactic characterization of the relation between two theories that have equivalent categories of models naturally in any Grothendieck topos. [ABSTRACT FROM AUTHOR]

In this paper we develop a notion of measure theory over boolean toposes reminiscent of the theory of von Neumann algebras. This is part of a larger project to study relations between topos theory and noncommutative geometry. The main result is a topos theoretic version of the modular time evolution of von Neumann algebras which take the form of a canonical $\mathbb{R}^{>0}$-principal bundle over any integrable locally separated boolean topos. [ABSTRACT FROM AUTHOR]

Topos theory is a category-theoretical axiomatization of set theory. Model categories are a category-theoretical framework for abstract homotopy theory. They are complete and cocomplete categories endowed with three classes of morphisms (called fibrations, cofibrations and weak equivalences) satisfying certain axioms. We would like to present an abstract framework for classical differential geometry as an extension of topos theory, hopefully comparable with model categories for homotopy theory. Functors from the category ... of Weil algebras to the category Sets of sets are called Weil spaces by Wolfgang Bertram and form the Weil topos after Eduardo J. Dubuc. The Weil topos is endowed intrinsically with the Dubuc functor, a functor from a larger category ... of cahiers algebras to the Weil topos standing for the incarnation of each algebraic entity of ... in the Weil topos. The Weil functor and the canonical ring object are to be defined in terms of the Dubuc functor. The principal objective of this paper is to present a category-theoretical axiomatization of the Weil topos with the Dubuc functor intended to be an adequate framework for axiomatic classical differential geometry. We will give an appropriate formulation and a rather complete proof of a generalization of the familiar and desired fact that the tangent space of a microlinear Weil space is a module over the canonical ring object. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra (Theory Appl Categ 24(7):117-147, 2010). For the topos of sets, we show that torsion-free functors on Loganathan's category L( S) of an inverse semigroup S are equivalent to a special class of non-strict representations of S, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of S. We describe the correspondence between directed and pullback preserving functors on L( S) and transitive and effective representations of S, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation, or, equivalently, an S-torsor, of an inverse semigroup S in the topos of sheaves $${\mathsf {Sh}}(X)$$ on a topological space X. We prove that the category of filtered functors from L( S) to the topos $${\mathsf {Sh}}(X)$$ is equivalent to the category of universal representations of S in $${\mathsf {Sh}}(X)$$ . We finally propose a definition of an inverse semigroup action in an arbitrary Grothendieck topos, which arises from a functor on L( S). [ABSTRACT FROM AUTHOR]

After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed ∞-categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every∞-topos has a hierarchy of "universal" univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying (n - 2)-truncated maps.We show that univalent families are preserved (and detected) by right adjoints to locally cartesian localizations, and use this to exhibit certain canonical univalent families in ∞-quasitopoi (certain ∞-categories of "separated presheaves", introduced here). We also exhibit some more exotic examples of univalent families, illustrating that a univalent family in an n-topos need not be (n - 2)-truncated, as well as some univalent families in the Morel-Voevodsky ∞-category of motivic spaces, an instance of a locally cartesian closed ∞-category which is not an n-topos for any 0 ≤ n ≤ ∞. Lastly, we show that any presentable locally cartesian closed ∞-category is modeled by a combinatorial type-theoretic model category, and conversely that the ∞-category underlying a combinatorial type-theoretic model category is presentable and locally cartesian closed. Under this correspondence, univalent families in presentable locally cartesian closed ∞-categories correspond to univalent fibrations in combinatorial type-theoretic model categories. [ABSTRACT FROM AUTHOR]

In the framework of the topos approach to quantum mechanics a kind of global valuation is introduced and studied. It allows us to represent certain features related to the logical consequences of properties about quantum systems when its phase space is endowed with an intuitionistic structure. [ABSTRACT FROM AUTHOR]

Until now, there is no explicit clarity about the question whether information, as similar to energy and mass, can be visualized as fundamental constituent of this universe or not, a question which is presently disputed. We follow here the line of argument which follows the first alternative, and give reasons for this by starting in terms of a metaphysical conception. [ABSTRACT FROM AUTHOR]

In this paper we deal with quantum theories on presheaves and sheaves on context categories consisting of commutative von Neumann algebras of bounded operators on a Hilbert space. Our aim is first to reduce presheaf-based topos quantum theory via sheafification and then to import quantum probabilities to the reduced sheaf quantum theory. The first is done by means of a functor that selects some expedient contexts. Note that since the functor defines a Grothendieck topology on the category consisting of all contexts, it induces a sheaf topos on which we construct a downsized quantum theory. We also show that the sheaf quantum theory can be replaced by a more manageable presheaf quantum theory. Quantum probabilities are imported by means of a Grothendieck topology that is defined on a category consisting of probabilities and that enables to regard them as intuitionistic truth-values. From these topologies, we construct another Grothendieck topology that is defined on the product of the context category and the probability category. It reflects the selection of contexts and the identification of probabilities with truth-values. We construct a quantum theory equipped with quantum probabilities as truth-values on the sheaf topos induced by the Grothendieck topology. [ABSTRACT FROM AUTHOR]

Connections between certain types of categories (logoses and toposes) and intuitionistic predicate logic was established in 1960-1970 by Lowvere. The possibility of extending this connection to some types of modal logics by using the internal structure of categories of particular type (logos) was also established. Category-theoretical constructs were hence used as one of the possible semantic interpretations of intuitionistic logic. This interpretation has also included intuionistic modal logics using different semantical tools such as adjoint pair of functors. In this paper, we discuss one of the possible extension of intuitionistic logic. [ABSTRACT FROM AUTHOR]

We generalize the Dubuc–Poveda representation theorem for MV-algebras so that it applies to other algebraic categories of residuated join-semilattices. In particular, as a corollary, we obtain a representation result for pre-linear residuated join-semilattices in terms of totally ordered fibers. The main result is analogous to the Zariski representation of (commutative) rings and it is proved using tools from topos theory. [ABSTRACT FROM AUTHOR]

We consider locales B as algebras in the tensor category sℓ of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections shB →ε in Galois theory and a Tannakian recognition theorem over si for the sℓ-functor Rel(E)... Rel(shB) ≌ (B-Mod) 0 into the sℓ-category of discrete B-modules. Thus, a new Tannaka recognition theorem is obtained, essentially different from those known so far. This equivalence follows from two independent results. We develop an explicit construction of the localic groupoid G associated by Joyal-Tierney to q, and do an exhaustive comparison with the Deligne Tannakian construction of the Hopf algebroid L associated to Rel(q*), and show they are isomorphic, that is, L ≌ O(G). On the other hand, we show that the sℓ-category of relations of the classifying topos of any localic groupoid G, is equivalent to the sℓ-category of L-comodules with discrete subjacent B-module, where L ≌ O(G). [ABSTRACT FROM AUTHOR]

We show that every abstract Krivine structure in the sense of Streicher can be obtained, up to equivalence of the resulting tripos, from a filtered opca (A,A') and a subobject of 1 in the relative realizability topos RT(A',A); the topos is always a Boolean subtopos of RT(A',A). We exhibit a range of non-localic Boolean subtriposes of the Kleene-Vesley tripos. [ABSTRACT FROM AUTHOR]

We establish, generalizing Di Nola and Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element. [ABSTRACT FROM AUTHOR]

Some categorical conditions are given that are sufficient to show that an enriched monad with a strength is a double exponential monad. The conditions hold for the double power locale monad (enriched over posets) and so as an application it is shown that the double power locale monad is a double exponential monad. A benefit is that this result about the double power locale monad can be established without the need for any detailed discussion of frame presentations or topos theory. [ABSTRACT FROM AUTHOR]

For a coherent site we construct a canonically associated enlarged coherent site, such that cohomology of bounded below complexes is preserved by the enlargement. In the topos associated to the enlarged site transfinite compositions of epimorphisms are epimorphisms and a weak analog of the concept of the algebraic closure exists. The construction is a variant of the work of Bhatt and Scholze on the pro-étale topology. [ABSTRACT FROM AUTHOR]

We investigate the semi-ringed topos obtained from the arithmetic site A of [3,4] , by extension of scalars from the smallest Boolean semifield B to the tropical semifield R + max . The obtained site [ 0 , ∞ ) ⋊ N × is the semi-direct product of the Euclidean half-line and the monoid N × of positive integers acting by multiplication. Its points are the same as the points A ( R + max ) of A over R + max and form the quotient of the adele class space of Q by the action of the maximal compact subgroup Z ˆ ⁎ of the idèle class group. The structure sheaf of the scaling topos endows it with a natural structure of tropical curve over the topos N × ˆ . The restriction of this structure to the periodic orbits of the scaling flow gives, for each prime p , an analogue C p of an elliptic curve whose Jacobian is Z / ( p − 1 ) Z . The Riemann–Roch formula holds on C p and involves real-valued dimensions and real degrees for divisors. [ABSTRACT FROM AUTHOR]

The theory of principal bundles makes sense in any $$\infty $$ -topos, such as the $$\infty $$ -topos of topological, of smooth, or of otherwise geometric $$\infty $$ -groupoids/ $$\infty $$ -stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure $$\infty $$ -group $$G$$ these $$G$$ - principal $$\infty $$ - bundles reproduce various higher structures that have been considered in the literature and further generalize these to a full geometric model for twisted higher nonabelian sheaf cohomology. We discuss here this general abstract theory of principal $$\infty $$ -bundles, observing that it is intimately related to the axioms that characterize $$\infty $$ -toposes. A central result is a natural equivalence between principal $$\infty $$ -bundles and intrinsic nonabelian cocycles, implying the classification of principal $$\infty $$ -bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber $$\infty $$ -bundles associated to principal $$\infty $$ -bundles subsumes a theory of $$\infty $$ - gerbes and of twisted $$\infty $$ - bundles, with twists deriving from local coefficient $$\infty $$ - bundles, which we define, relate to extensions of principal $$\infty $$ -bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice $$\infty $$ -topos. [ABSTRACT FROM AUTHOR]

In [8] Joyal and Moerdijk have shown that realizability toposes over partial combinatory algebras (pca) host classes of small maps giving rise to initial ZF -algebras providing models of intuitionistic Zermelo–Fraenkel set theory IZF . Here we show that this can be extended to a much wider class of realizability toposes as considered in [4] and [3] . For this purpose we first show this result for nested realizability toposes RT ( A , A # ) induced by a pca A together with a sub-pca A # as considered implicitly in [4] and then show that it is preserved by restriction to subtoposes. This suffices since all toposes considered in [4] and [3] arise as subtoposes of some nested realizability toposes. [ABSTRACT FROM AUTHOR]

We show that in locally-ringed connected topoi the primes dividing the period and index of a Brauer class coincide. The result applies in particular to Brauer classes on connected schemes, algebraic stacks, topological spaces and to the projective representation theory of profinite groups. [ABSTRACT FROM AUTHOR]

The ways the relationship between theory and practice is understood and implemented in study programmes is a question that has been posed since antiquity, and that the Bologna process and the reform of the professional bachelor programme problem have intensified also in Denmark. Professional bachelor programmes have become increasingly abstract, and the individual educational institutions have been merged into university colleges [Undervisningsministeriet (2007). Bekendtgørelse om uddannelsen til professionsbachelor som lærer i folkeskolen. BEK nr 219 af 12/03/2007. Ministry of Education. Proclamation of Professional Bachelor Teacher Education Programme. Retrieved May 2014 from, https://www.retsinformation.dk/Forms/R0710.aspx?id=25302]. In this article, we reflect on the epistemological challenges we faced during a research project on the correlation between theory and practice and drop-out rates at four professional bachelor programmes in Denmark. Our point of departure was to view theory and practice as across time and place. But the contrasts among the four programmes made it clear to us that the various understandings of the relations between theory and practice that we observed had to be seen as mixed cultural practices undergoing a process of change. The concepts of theory and practice, the places, the academic content and teaching methods, and knowledge forms we observed were mutually interdependent, and fought for recognition as central to the way in which professional bachelor programmes were reproduced and transformed. In this article, we use the comprehensive transformation processes of professional bachelor programmes as reasons for reframing/restating questions regarding theory and practice in which theory and practice are viewed as cultural forms in education and professional practice. We suggest an actual didactics of theory--practice as part of professional bachelor programme practice for lecturers, students and supervisors. [ABSTRACT FROM AUTHOR]

Homotopy type theory may be seen as an internal language for the ∞-category of weak ∞-groupoids. Moreover, weak ∞-groupoids model the univalence axiom. Voevodsky proposes this (language for) weak ∞-groupoids as a new foundation for Mathematics called the univalent foundations. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those ‘discrete’ groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of ‘elementary’ of ∞-toposes. We prove that sets in homotopy type theory form a ΠW-pretopos. This is similar to the fact that the 0-truncation of an ∞-topos is a topos. We show that both a subobject classifier and a 0-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover the 0-object classifier for sets is a function between 1-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets. [ABSTRACT FROM AUTHOR]

We define a notion of homotopy in the effective topos. [ABSTRACT FROM AUTHOR]

We report first steps towards elucidating the relationship between contextuality, measurementbased quantum computation (MBQC) and the non-classical logic of a topos associated with the computation. We show that, in a class of MBQC, classical universality requires non-classical logic, which is 'consumed' during the course of the computation, thereby pinpointing another potential quantum computational resource. [ABSTRACT FROM AUTHOR]

This article presents results from a qualitative pilot survey of science, technology, engineering, and math (STEM) researchers concerning techniques used to create graphics for research articles. The survey aimed to induce a methodological vocabulary for a larger project designed to describe and improve STEM visual literacy for nonexperts. However, the survey also revealed interesting problems for investigation—chief among them a mismatch between STEM visual pedagogy and praxis. In addition, participants supplied a handlist of STEM visual communication texts that have informed their praxis. Survey results are presented in the form of a topology—a frequency-based representation of the topics framing participants’ discussion of STEM visual invention. [ABSTRACT FROM PUBLISHER]

A theorem of A.M.Pitts (1986) states that essential surjections of toposes bounded over a base topos S are of effective lax descent. The symmetric monad M on the 2-category of toposes bounded over S is a KZ-monad (Bunge-Carboni 1995) and the M-maps are precisely the S-essential geometric morphisms (Bunge-Funk 2006). These last two results led me to conjecture and then prove the general lax descent theorem that is the subject matter of this paper.By a 'Pitts KZ-monad' on a 2-category K it is meant here a locally fully faithful equivariant KZ-monad M on K that is required to satisfy an analogue of Pitts' theorem on bicomma squares along essential geometric morphisms. The main result of this paper states that, for a Pitts KZ-monad M on a 2-category K ('of spaces'), every surjective M-map is of effective lax descent. There is a dual version of this theorem for a Pitts co-KZ-monad N. These theorems have (known and new) consequences regarding (lax) descent for morphisms of toposes and locales. [ABSTRACT FROM AUTHOR]

We show that sheaves on Grothendieck quantales, which are the quantales that correspond to Grothendieck toposes, may be defined in several equivalent ways. The main topic of this paper is the characterization of internal posets and suplattices in the topos of sheaves on a Grothendieck quantale Q , in terms of categories enriched in a suitable quantaloid and by means of Q -modules. We thereby extend the existing results concerning locales. [ABSTRACT FROM AUTHOR]

Let [ABSTRACT FROM AUTHOR]

Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the computationally dense ones) are seen to be the ones whose 'lifts' to a kind of completion have right adjoints. We characterize topos inclusions corresponding to a general form of relative computability. We characterize pcas whose realizability topos admits a geometric morphism to the effective topos. [ABSTRACT FROM AUTHOR]

In this paper, we construct a sheaf-based topos quantum theory. It is well known that a topos quantum theory can be constructed on the topos of presheaves on the category of commutative von Neumann algebras of bounded operators on a Hilbert space. Also, it is already known that quantization naturally induces a Lawvere-Tierney topology on the presheaf topos. We show that a topos quantum theory akin to the presheaf-based one can be constructed on sheaves defined by the quantization-induced Lawvere-Tierney topology. That is, starting from the spectral sheaf as a state space of a given quantum system, we construct sheaf-based expressions of physical propositions and truth objects, and thereby give a method of truth-value assignment to the propositions. Furthermore, we clarify the relationship to the presheaf-based quantum theory. We give translation rules between the sheaf-based ingredients and the corresponding presheaf-based ones. The translation rules have "coarse-graining" effects on the spaces of the presheaf-based ingredients; a lot of different proposition presheaves, truth presheaves, and presheaf-based truth-values are translated to a proposition sheaf, a truth sheaf, and a sheaf-based truth-value, respectively. We examine the extent of the coarse-graining made by translation. [ABSTRACT FROM AUTHOR]