arXiv subjects

Eduardo J. Dubuc

Publications and source records attributed to Eduardo J. Dubuc.

At least 19 recordsLinked to original sources

arXiv2023

From Yoneda to Topoi morphisms

In this note we show how two fundamental results in Topos theory follow by repeated use of Yoneda's Lemma, the formalism of natural transformations and very basic category theory. In Lemma 9.4, we show the fundamental result SGA4 EXPOSE IV Proposition 4.9.4, which says that for any site $\cc{C}$, the canonical functor $\cc{C} \mr{\varepsilon} Sh(\cc{C})$ into the category of sheaves, classifies sites morphisms $\cc{C} \mr{} \cc{Z}$ into any topos $\cc{Z}$. After the usual Yoneda's Lemma, Lemma 3.1, we show that the Yoneda functor $\cc{C} \mr{h} \Eop{C}$ classifies functors $\cc{C} \mr{} \cc{Z}$ into any cocomplete category $\cc{Z}$, via a cocontinuos extension $\Eop{C} \mr{} \cc{Z}$, Lemma 5.2. Then we reach Lemma 9.4 by an step by step enrichment of 5.2. All we use is Yoneda's Lemma, over and over again, and the Yoga of natural transformations. In Lemma 10.1 we show the equivalence between \emph{flatness} and \emph{left exactness} for functors from finitely complete categories into any topos. Our proof is elementary, we show how basic exactness properties of sets prove the result for set valued functors, then we generalize to functors valued in any topos utilizing results of the previous sections and the Yoga of natural transformations. Besides the thread we follow,nothing here is new, although we haven't seen 10.1 proved this way before.

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arXiv2023

The 2-localization of a model category

In this paper we study a 2-dimensional version of Quillen's homotopy category construction. Given a category $\mathscr{A}$ and a class of morphisms $Σ\subset \mathscr{A}$ containing the identities, we construct a 2-category $\mathcal{H}o(\mathscr{A})$ obtained by the addition of 2-cells determined by homotopies. A salient feature here is the use of a novel notion of cylinder introduced in \cite{e.d.2}. The inclusion 2-functor $\mathscr{C} \longrightarrow \mathcal{H}o(\mathscr{A})$ has a universal property which implies that it will be the 2-localization of $\mathscr{A}$ at $Σ$ as soon as the arrows of $Σ$ become equivalences in $\mathcal{H}o(\mathscr{A})$. This is then used to obtain 2-localizations of a model category $\mathscr{A}{C}$, with $Σ= \mathcal{W}$, the weak equivalences, and $\mathscr{A} = \mathscr{C}_{fc}$, the full subcategory of fibrant-cofibrant objects, as well as with $\mathscr{A} = \mathscr{C}$. The set of connected components of the hom categories yields Quillen's results. We follow the general lines established in \cite{e.d.2}, \cite{e.d.} for model bicategories. The development here is not just the examination of the general theory in a particular case. It is not concerned with and avoids the problems which arise when dealing with non invertible 2-cells. Also, the use here of functorial factorization adds further simplifications by eliminating the need of pseudofunctors. New proofs are produced which are not a mere simplified adaptation of the ones of the general case.

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arXiv2021

A construction of 2-filtered bicolimits of categories

We define the notion of 2-filtered 2-category and give an explicit construction of the bicolimit of a category valued 2-functor. A category considered as a trivial 2-category is 2-filtered if and only if it is a filtered category, and our construction yields a category equivalent to the category resulting from the usual construction of filtered colimits of categories. Weaker axioms suffice for this construction, and we call the corresponding notion pre 2-filtered 2-category. The full set of axioms is necessary to prove that 2-filtered bicolimits have the properties corresponding to the essential properties of filtered bicolimits. Kennison already considered filterness conditions on a 2-category under the name of bifiltered 2-category. It is easy to check that a bifiltered 2-category is 2-filtered, so our results apply to bifiltered 2-categories. Actually Kennison's notion is equivalent to ours, but the other direction of this equivalence is not entirely trivial.

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arXiv2017

Spans and simplicial families

In this paper we consider simplicial families, that is, simplicial objects indexed by a simplicial set. We develop a method to construct family hypercover refinements of a cover family based on the notion of \emph{n-spans} that we introduce here. In [The fundamental progroupoid of a general topos, Journal of Pure and Applied Algebra 212 (2008)] we had introduced the notion of \emph{covering projection} in a topos. They are locally constant objects satisfying an additional condition which is valid in all locally constant objects when the topos is locally connected, and developed the theory of the fundamental groupoid of a general topos. Here we show that covering projections can be obtained as objects constructed from a descent datum of a simplicial set on a family of sets. We construct a groupoid $\nn{G}_\cc{H}$ such that the category $\cc{G}_\cc{H}$ of covering projections trivialized by $\cc{H}$ is its classifying topos. This determines a protopos $\{\cc{G} \cc{H}\}_\cc{H}$ and a progroupoid $\{\nn{G}_\cc{H}\}_\cc{H}$, suitable indexed by a filtered poset of hypercovers. Then we show that this progroupoid classifies torsors. This construction is novel also in the case of locally connected topoi, showing that locally constant object in a locally connected topos are constructed by descent from a descent datum on a family of sets. The salient feature that distinguishes locally connected topoi is that the progroupoid is \emph{strict}, that is, the transition morphisms are surjective on triangles, or, equivalently, the transition inverse image functors in the underlying indcategory are full and faithful.

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arXiv2017

Some remarks on infinitesimals in MV-algebras

Replacing $\{0\}$ by the whole ideal of infinitesimals yields a weaker notion of \emph{archimedean element} that we call \emph{quasiarchimedean}. It is known that semisimple MV-algebras with compact maximal spectrum (in the co-Zarisky topology) are exactly the hyperarchimedean algebras. We characterise all the algebras with compact maximal spectrum as being \emph{quasihyperarchimedean} \mbox{MV-algebras,} which in a sense are non semisimple hyperarchimedean algebras. We develop some basic facts in the theory of MV-algebras along the lines of algebraic geometry, where infinitesimals play the role of nilpotent elements, and prove a MV-algebra version of Hilbert's Nullstellensatz. Finally we consider the relations (some inedited) between several elementary classes of MV-algebras in terms of the ideals that characterise them, and present elementary (first order with denumerable disjunctions) proofs in place of the \mbox{set-theoretical} usually found in the literature.

math.LO↗

arXiv2015

Tannaka Theory for Topos

We consider locales $B$ as algebras in the tensor category $s\ell$ of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections $sh(B) \stackrel{q}{\longrightarrow} \mathcal{E}$ in Galois theory [An extension of the Galois Theory of Grothendieck, AMS Memoirs 151] and a Tannakian recognition theorem over $s\ell$ for the $s\ell$-functor $Rel(E) \stackrel{Rel(q^*)}{\longrightarrow} Rel(sh(B)) \cong (B$-$Mod)_0$ into the $s\ell$-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, $L \cong \mathcal{O}(G)$. On the other hand, we show that the $s\ell$-category of relations of the classifying topos of any localic groupoid $G$, is equivalent to the $s\ell$-category of $L$-comodules with discrete subjacent $B$-module, where $L = \mathcal{O}(G)$. We are forced to work over an arbitrary base topos because, contrary to the neutral case developed over Sets in [A Tannakian Context for Galois Theory, Advances in Mathematics 234], here change of base techniques are unavoidable.

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arXiv2014

On the equivalence between MV-algebras and $l$-groups with strong unit

In "A new proof of the completeness of the Lukasiewicz axioms"} (Transactions of the American Mathematical Society, 88) C.C. Chang proved that any totally ordered $MV$-algebra $A$ was isomorphic to the segment $A \cong Γ(A^*, u)$ of a totally ordered $l$-group with strong unit $A^*$. This was done by the simple intuitive idea of putting denumerable copies of $A$ on top of each other (indexed by the integers). Moreover, he also show that any such group $G$ can be recovered from its segment since $G \cong Γ(G, u)^*$, establishing an equivalence of categories. In "Interpretation of AF $C^*$-algebras in Lukasiewicz sentential calculus" (J. Funct. Anal. Vol. 65) D. Mundici extended this result to arbitrary $MV$-algebras and $l$-groups with strong unit. He takes the representation of $A$ as a sub-direct product of chains $A_i$, and observes that $A \overset {} {\hookrightarrow} \prod_i G_i$ where $G_i = A_i^*$. Then he let $A^*$ be the $l$-subgroup generated by $A$ inside $\prod_i G_i$. He proves that this idea works, and establish an equivalence of categories in a rather elaborate way by means of his concept of good sequences and its complicated arithmetics. In this note, essentially self-contained except for Chang's result, we give a simple proof of this equivalence taking advantage directly of the arithmetics of the the product $l$-group $\prod_i G_i$, avoiding entirely the notion of good sequence.

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arXiv2014

A theory of 2-pro-objects (with expanded proofs)

Grothendieck develops the theory of pro-objects over a category $\mathsf{C}$. The fundamental property of the category $\mathsf{Pro}(\mathsf{C})$ is that there is an embedding $\mathsf{C} \overset{c}{\longrightarrow} \mathsf{Pro}(\mathsf{C})$, the category $\mathsf{Pro}(\mathsf{C})$ is closed under small cofiltered limits, and these limits are free in the sense that for any category $\mathsf{E}$ closed under small cofiltered limits, pre-composition with $c$ determines an equivalence of categories $\mathcal{C}at(\mathsf{Pro}(\mathsf{C}),\,\mathsf{E})_+ \simeq \mathcal{C}at(\mathsf{C},\, \mathsf{E})$, (where the "$+$" indicates the full subcategory of the functors preserving cofiltered limits). In this paper we develop a 2-dimensional theory of pro-objects. Given a 2-category $\mathcal{C}$, we define the 2-category $2\hbox{-}\mathcal{P}ro(\mathcal{C})$ whose objects we call 2-pro-objects. We prove that $2\hbox{-}\mathcal{P}ro(\mathcal{C})$ has all the expected basic properties adequately relativized to the 2-categorical setting, including the universal property corresponding to the one described above. We have at hand the results of $\mathcal{C}at$-enriched category theory, but our theory goes beyond the $\mathcal{C}at$-enriched case since we consider the non strict notion of pseudo-limit, which is usually that of practical interest.

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arXiv2014

Categorías. Los 30 primeros años

This Note describe my own recollection of the first 30 years of Category Theory, it is not the result of any historical investigation. The choice of concepts and its evaluation is my own, necessarily subjective. It follows a chronological line along the 21 references in the bibliography. It is a faithful version, based on 10 manuscript transparencies utilized at a conference given in October 2013 at the fourth ENHEM (Escuela Nacional de Historia y Educacìon Matemàtica) congress in Cali, Colombia. As such, its content is bounded by time, and it does not include anything that was not said there. Text is in spanish.

math.HO↗

arXiv2012

A Tannakian Context for Galois

Strong similarities have been long observed between the Galois (Categories Galoisiennes) and the Tannaka (Categories Tannakiennes) theories of representation of groups. In this paper we construct an explicit (neutral) Tannakian context for the Galois theory of atomic topoi, and prove the equivalence between its fundamental theorems. Since the theorem is known for the Galois context, this yields, in particular, a proof of the fundamental (recognition) theorem for a new Tannakian context. This example is different from the additive cases or their generalization, where the theorem is known to hold, and where the unit of the tensor product is always an object of finite presentation, which is not the case in our context.

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arXiv2011

A Simple Proof of McNaughton Theorem

In this note we exhibit a very simple proof of McNaughton Theorem, almost right out of the definitions, and at the same time we observe that this theorem does not depend of Chang's completeness theorem.

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arXiv2011

A construction of 2-cofiltered bilimits of topoi

We show the existence of bilimits of 2-cofiltered diagrams of topoi, generalizing the construction of cofiltered bilimits developed in "SGA 4 Springer LNM 270 (1972)". For any given such diagram, we show that it can be represented by a 2-cofiltered diagram of small sites with finite limits, and we construct a small site for the inverse limit topos. This is done by taking the 2-filtered bicolimit of the underlying categories and inverse image functors. We use the construction of this bicolimit developed in "A construction of 2-filtered bicolimits of categories, Cah. Top. et Geo. Diff. Vol. XLVII-2 (2006)", where it is proved that if the categories in the diagram have finite limits and the transition functors are exact, then the bicolimit category has finite limits and the pseudocone functors are exact. An application of our result here is the fact that every Galois topos has points "2-Filteredness and the point of every Galois topos, Proc. CT2007, App. Cat. St., Vol. 18, 2, (2010)".

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arXiv2008

Representation theory of mv-algebras

In this paper we develop a general representation theory for mv-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of mv-algebras and mv-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. \emph{We prove that any mv-algebra is isomorphic to the mv-algebra of all global sections of a sheaf of mv-chains on a compact topological space}. This result is intimately related to McNaughton's theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton's. On spite of the language utilized in this abstract, we wrote this paper in a way that, we hope, could be read without much acquaintance with either sheaf theory or mv-algebra theory.

math.LO↗

arXiv2007

2-filteredness and the point of every Galois topos

A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bi-limits of topoi, we show that every Galois topos has a point.

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arXiv2007

The fundamental progroupoid of a general topos

It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the \emph{fundamental progroupoid}, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and can not be replaced by a localic groupoid. The classifying topos in not any more a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver July 2004.

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arXiv2006

Quasitopoi over a base category

In this paper we develop the theory of quasispaces (for a Grothendieck topology) and of concrete quasitopoi, over a suitable base category. We introduce the notion of f-regular category and of f-regular functor. The f-regular categories are regular categories in which every family with a common codomain can be factorized into a strict epimorphic family followed by a (single) monomorphism. The f-regular functors are (essentially) functors that preserve finite strict monomorphic and arbitrary strict epimorphic families. These two concepts furnish the context to develop the constructions of the theory of concrete quasitopoi over a suitable base category, which is a theory of pointed quasitopoi. Our results on quasispaces and quasitopoi, or closely related ones, were already established by Penon, but we prove them here with different assumptions and generality, and under a completely different light.

math.CT↗

arXiv2006

Topological functors as familiarly-fibrations

In this paper we develop the theory of topological categories over a base category, that is, a theory of topological functors. Our notion of topological functor is similar to (but not the same) the existing notions in the literature (see Borceux "Handbook of categorical algebra 2, 7.3"), and it aims at the same examples. In our sense, a (pre) topological functor is a functor that creates cartesian families. A topological functor is, in particular, a fibration, and our emphasis is put in this fact.

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arXiv2003

On the representation theory of Galois and Atomic Topoi

We elaborate on the representation theorems of topoi as topoi of discrete actions of various kinds of localic groups and groupoids. We introduce the concept of "proessential point" and use it to give a new characterization of pointed Galois topoi. We establish a hierarchy of connected topoi: [1. essentially pointed Atomic = locally simply connected], [2. proessentially pointed Atomic = pointed Galois], [3. pointed Atomic]. These topoi are the classifying topos of, respectively: 1. discrete groups, 2. prodiscrete localic groups, and 3. general localic groups. We analyze also the unpoited version, and show that for a Galois topos, may be pointless, the corresponding groupoid can also be considered, in a sense, the groupoid of "points". In the unpointed theories, these topoi classify, respectively: 1. connected discrete groupoids, 2. connected (may be pointless) prodiscrete localic groupoids, and 3. connected groupoids with discrete space of objects and general localic spaces of hom-sets, when the topos has points (we do not know the class of localic groupoids that correspond to pointless connected atomic topoi). We comment and develop on Grothendieck's galois theory and its generalization by Joyal-Tierney, and work by other authors on these theories.

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