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Yoneda embedding and finite limit statements


Pierre-Alain Jacqmin
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Please, excuse me for this naive question, but I am new to the subject. One consequence of Yoneda embedding is that one can translate proofs and constructions about finite limits from the category of sets to any finitely complete category. If we considered  \mathbb{T}_1 the finite limit theory of categories with chosen finite limits and  \mathbb{T}_2 the theory whose category of models in a topos E is E itself, can we "transfer" such proofs and constructions from  \mathbb{T}_1 to  \mathbb{T}_2 via the technique of bridges and prove the above mentioned consequence of the Yoneda embedding? If this is possible, the immediate follow up question will be: Is it possible to do the same with other kinds of embedding theorems such as Lubkin-Freyd-Mitchell embedding theorem for abelian categories. And is it possible to do the same with other situations where an embedding theorem does not exist (yet) ?

Thank you in advance for your answer,

Pierre-Alain


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L.Lafforgue
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@pierre-alain-jacqmin

To any small category with chosen finite limits, one can associate the set of objects of this category. More generally, for any topos E, to any inner category with chosen finite limits in E one can associate the object of E which classifies "objects" of this inner category. This defines a morphism from the classifying topos of the theory of catgories with chosen finite limits to the classifying topos of the theory of objects. But this morphism is not at all an equivalence : the two theories don't have the same set-theoretic models or, equivalently, their classifying toposes do not have the same points.

As a side remark, the theory of categories with chosen finite limits is cartesian, so its classifying topos is a presheaf topos. Of course, the theory of objects is also a presheaf type theory.

On the other hand, the canonical functor l (defined as the composite of the Yoneda embedding y with the sheafification functor) of an (essentially) small category C endowed with a Grothendieck topology J into the topos E of sheaves on the site (C,J) can be seen as inducing an equivalence of toposes between the category of J-sheaves on C and the category of sheaves on E for the canonical topology (defined by deciding that a family of morphisms to an object of E is a covering family iff it is globally epimorphic). Limis or colimits are well-defined in any topos and are invariant under equivalences. So we can consider these invariant notions and compare how they are expressed in the language of E and in the language of C endowed with the topology J. In the case of colimits, this comparison is not trivial and provides a very natural notion of "good" colimit in a category C, especially if J is the canonical topology of C or, more generally, a sub-canonical topology. This is an example of topos-theoretic bridge.

I had given this concrete expression of topos-theoretic colimits in the last part of my lectures at the "Toposes in Como" summer school :


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Pierre-Alain Jacqmin
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Many thanks for your answer. I now have a better understanding of the technique of bridges. I found the lecture very interesting, especially the last part on colimits.


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Olivia Caramello
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@pierre-alain-jacqmin

Welcome to the forum! Concerning your follow up question, you may want to investigate more generally how topos-theoretic invariants can be transferred to site-theoretic properties or notions along functors which induce (when regarded as morphisms or comorphisms of sites in a suitable way) an equivalence of toposes. Such functors are systematically investigated in this work (as observed by @l-lafforgue in his message and explained there, the Yoneda embedding is one example of this kind of functors).

 


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Pierre-Alain Jacqmin
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@ocaramello

Thank you for your answer. I will have a look to your paper. I think I will start by looking at explicit examples such as the Yoneda embedding since, as described below, it seems to nicely link with my research.

 

@l-lafforgue

I like the idea you explain towards the end of the video, i.e., to compute a colimit of a diagram in a category  \mathbb{C} in the topos E generated by the canonical topology on  \mathbb{C} . There is a similar approach which is to compute the colimit in its pro-completion (i.e. the dual of the ind-completion) given by  \mathsf{Lex}(\mathbb{C},\mathsf{Set})^{\mathsf{op}} where  \mathsf{Lex}(\mathbb{C},\mathsf{Set}) denotes the category of finite limit preserving functors  \mathbb{C} \to \mathsf{Set}. The common features and differences I can immediately notice between the two approaches are:

- In both cases,  \mathbb{C} fully faithfully embeds in the "completion".

- Both completions are complete and cocomplete, so that the colimit always exists (even if it does not in  \mathbb{C} ).

- Both completions enjoy other good exactness properties, independently of  \mathbb{C} : the dual of the pro-completion is a locally finitely presentable category (in other words the pro-completion is co-essentially algebraic). The topos E has dual properties: it is locally presentable and moreover it is a topos.

- The embedding of   \mathbb{C} into its pro-completion preserves existing colimits, so the notion of colimits there is not different from the one in  \mathbb{C} if it exists. Besides, the embedding of  \mathbb{C} in E does not preserve colimits and, as you explain in your talk, the colimit in E is in some sense better than the colimit in  \mathbb{C} (even if it exists).

- The pro-completion inherits many properties from  \mathbb{C} (see this work for a general theorem generalizing many existing ones). As I understand from @ocaramello 's answer, such transfer properties are also studied in the context of E.

- The stability property I mentioned in my previous item can be used in a similar way as the 'toposes as bridges technique'  developed by @ocaramello (at least as far as I understand it) (see e.g. this work).

I believe it would be interesting to compare these two approaches and deeper understand their links.


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L.Lafforgue
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@pierre-alain-jacqmin

Your comment draws attention to a very intriguing paradox :

On the one hand, the notion of category is symmetrical, in the sense that the opposite of a category is a category, with the consequence that any notion or construction formulated in the language of categories admits a dual notion or construction. In particular, the notions of limits and colimits are dual, as are the notions of ind-completions and pro-completions, etc.

On the other hand, when one considers a concrete category, replacing it by the opposite category most often completely changes the point of view and, in particular, the constructions of limits and colimits in a given category most often do not have the same level of difficulty and depth at all.

For example, the opposite of the algebraic category C of commutative rings is the category of affine schemes to which all the usual vocabulary of geometry applies whereas it does not apply at all to C.

Generally speaking, when one category can be considered "geometric", the opposite category cannot be considered "geometric" at all.

In a "geometric" category, the limits that are well defined are usually relatively elementary, whereas the colimits are usually much more subtle, deeper and mathematically richer. For example, looking for quotients in algebraic geometry has given rise to Mumford's "Geometric Invariants Theory" (GIT) as well as the theory of stacks (algebraic or not) introduced by Grothendieck and then studied by Giraud, Artin, Deligne, Mumford and many others.

When a category can be seen as geometric, it is natural to consider one or several Grothendieck topologies on this category and to send it into the associated topos, made up of the sheaves on this category for the different topologies chosen. These embedding functors respect finite limits. On the other hand, they do not respect colimits and one obtains different notions of colimits according to the chosen topologies. This corresponds to the subtlety of the question of colimits constructions in geometric categories.

This fundamental asymmetry of geometry is reflected in particular by the fact that the opposite category of a topos (different from the trivial topos with a single object and a single morphism) is never a topos. In any topos, functors of fiber products with a given object on a given base respect arbitrary colimits, whereas amalgamated summation functors do not respect limits except if the topos is trivial. In particular, the opposite of the category Set is not a topos and, already in Set, the construction of colimits, i.e. quotient sets, is more subtle and richer than the construction of limits, i.e. subsets defined by equations.

On the other hand, the proper of linear properties is to be symmetrical: the opposite of an additive category is an additive category, and the opposite of an abelian category is an abelian category.

This means in particular that all cohomological functors, which are defined from "geometrical" categories to "linear" categories, go from a fundamentally asymmetrical world to a symmetrical world. Going to the symmetrical world of linear structures makes it possible for "dualities" to appear: Poincaré's duality and its reinterpretation and generalisation by Grothendieck in what he called the "formalism of the six operations" are at the centre of the linearisation of geometry by cohomological fonctors.


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Pierre-Alain Jacqmin
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@l-lafforgue

Thank you for your interesting comment. I would like to add one remark to it.

As you wrote, some categories can be thought to belong to the "geometric world" (or "topological world"); and limits in them are often easier to understand than colimits. But as we know, some other categories can be thought to belong to the "algebraic world". In those categories as well, limits are easier to understand than colimits. The reason is quite simple: objects in algebraic categories are equipped with algebraic operations, defined as functions  A^n \to A , and products interact well with limits (they commute with each other) but in general they don't commute with colimits.

However, the geometric world and the algebraic world are in some sense dual to each other. The dual of the category of topological spaces is (quasi)-algebraic, the dual of the category of affine scheme is the category of commutative rings, the dual of the category of locales is the category of frames, the dual of the category of compact Hausdorff spaces is the category of commutative unital C*-algebras, etc.

Since those two worlds are dual to each other, but in each of them limits are easier to understand than colimits, this means that the equivalences appearing in these dualities contain a lot of information. Moreover, the case of the category of sets which trivially belongs to both worlds is intriguing, but confirms that, intrinsically, limits are easier to understand than colimits. 


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L.Lafforgue
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@pierre-alain-jacqmin

In categories of set-based models of algebraic theories, the forgetful functors of the algebraic structures respect limits whereas, in general, they do not respect colimits. This is probably why you write that in these categories limits are easier to understand than colimits.

However, one can also consider that the natural way to present a set-based model M of an algebraic theory T (or more generally of a "presheaf type" theory T, i.e. a theory whose classifying topos is equivalent to a topos of presheaves) is in terms of "generators" and "relations" : such a model M is presented by a family G of generators g subject to a family R of relations r (formulated in the language of the theory T) if, for any model N of the theory T, giving a morphism M --> N is equivalent to giving a family of elements of N indexed by the elements g of G and which verifies the relations r of the family R.

As the presentations in terms of generators and relations of the models M of T consist in classifying the morphisms M --> N, they are very well adapted to the calculation of colimits. For any diagram D with values in the category of models of an algebraic theory (or more generally "presheaf type" theory) T, and where each object is described in terms of generators and relations, the colimit of this diagram admits an almost immediate description in terms of a family of generators and relations which is deduced from those from which one started.

On the other hand, the same cannot be said of limits.

Consider for example a ring A described in terms of generators and relationships and provided with an action of a group G. In general, we cannot deduce a description in terms of generators and relations of the subring of A consisting of the invariant elements under the action of G.

One of Hilbert's most famous theorems was to show that if A is a ring of polynomials in a finite number of variables with coefficients in some field, and if the action of G respects the graduation of A by the total degree of each monomial, then the subring of fixed points can be described by a finite number of generators and relations. However, Hilbert's demonstration does not allow at all to concretely exhibit such a finite family of generators and relations. That it is possible to demonstrate the existence of something without giving a concrete procedure for constructing such a thing was completely new in mathematics. This is why Hilbert's demonstration shocked mathematicians of his time. Gordan, who was one of the experts of invariants, commented : "This is theology, not mathematics!"

Hilbert's theorem was in fact the starting point for the "theory of invariants" which was taken up and globalised much later by Mumford in the context of scheme theory.

The historical importance of Hilbert's theorem illustrates how deep are the questions of limits of commutative rings diagrams, i.e. colimits of affine schemes diagrams. Mumford's theory has made it possible to go beyond the framework of affine schemes to consider colimits of diagrams of schemes. The development of this theory led Mumford to introduce and study the notions of "stability" and "semi-stability" which have spread to ever more diverse parts of mathematics.

 

ps : In my last post, I forgot to mention the obvious but most important fact that, for any field k, the category of finite dimensional vector spaces over k is self-dual : it is naturally equivalent to its opposite category. This was generalised by Grothendieck to the context of categories of linear sheaves. As these categories have non trivial local Ext cohomology, the abelian categories of linear sheaves have to be replaced by their associated derived categories to make possible some self-duality. Furthermore, the "finite dimension" condition has to be replaced by a requirement that the sheaves under consideration are "constructible" in some sense. This is part of Grothendieck's "six operations formalism".


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