{"id":4133,"date":"2021-09-22T20:09:19","date_gmt":"2021-09-22T18:09:19","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4133"},"modified":"2023-04-17T18:39:28","modified_gmt":"2023-04-17T16:39:28","slug":"topological-functors-i-definition-duality-limits-and-colimits","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4133","title":{"rendered":"Topological functors I: definition, duality, limits and colimits"},"content":{"rendered":"\n

I have already mentioned topological functors here<\/a>, only very quickly, where I mostly directed you to Sections 21 and 22 of [1]. Topological functors are one way of describing categorically what makes the category Top<\/strong> of topological spaces so special. However, there are many other topological functors in nature. The situation is a bit similar as the notion of ring in algebra, which is one way of describing Z<\/strong>\/n<\/em>Z<\/strong> abstractly, although there are many other rings in nature.<\/p>\n\n\n\n

Here is how I will proceed. I will explain what topological functors are by taking the example of Top<\/strong>. There are many other examples, such as preordered sets, prestreams, streams, or d-spaces. Before I come to them, I will list, and prove, a few useful properties of topological functors. I hope that this presentation will be easier to read than [1], which I find a bit demanding.<\/p>\n\n\n\n

Topological functors: the canonical example<\/h2>\n\n\n\n

All right. The starting point is to consider not Top<\/strong> itself, rather the forgetful functor U<\/em> : Top<\/strong> \u2192 Set<\/strong> that maps every topological space to its underlying set, and see what special properties U<\/em> has.<\/p>\n\n\n\n

Property 1: Faithfulness<\/h3>\n\n\n\n

One of those properties is that U<\/em> is faithful<\/em>. Given any two topological spaces X<\/em> and Y<\/em>, U<\/em> maps every morphism f<\/em> from X<\/em> to Y<\/em> in Top<\/strong> (a continuous map) to… simply f<\/em>, seen as a mere function. Hence U<\/em> is injective from the set Top<\/strong>(X<\/em>,Y<\/em>) of morphisms f<\/em> in Top<\/strong> from X<\/em> to Y<\/em>, to Set<\/strong>(U<\/em>(X<\/em>),U<\/em>(Y<\/em>)). Top<\/strong>(X<\/em>,Y<\/em>) and Set<\/strong>(U<\/em>(X<\/em>),U<\/em>(Y<\/em>)) are examples of homsets<\/em>, namely of sets of morphisms between two given objects. Hence U<\/em> is injective on homsets, but the standard name for that property is to say that U<\/em> is faithful<\/em>.<\/p>\n\n\n\n

Oh, U<\/em> is certainly not surjective on homsets in general: that would mean that every function between topological spaces is continuous, which is simply wrong.<\/p>\n\n\n\n

Property 2: every U<\/em>-source has a U<\/em>-initial lift (what gibberish! … but read on)<\/h3>\n\n\n\n

Let me call a family<\/em> of topological spaces any collection (Yi<\/sub><\/em>)i<\/em> \u2208 I<\/em><\/sub> of topological spaces Yi<\/sub><\/em> indexed by some class I<\/em>. In other words, a family is not a set, rather a map from I<\/em> to the class of topological spaces. Perhaps more importantly, families may be indexed by a proper class, not just by sets I<\/em>; and, in doing so, we are entitled to repeat several times the same space Yi<\/sub><\/em>.<\/p>\n\n\n\n

Given a set E<\/em>, any family of topological spaces (Yi<\/sub><\/em>)i<\/em> \u2208 I<\/em><\/sub>, and maps gi<\/sub><\/em> : E<\/em> \u2192 Yi<\/sub><\/em>, one for each i<\/em> in I<\/em>, there is a coarsest topology on E<\/em> that makes every gi<\/sub><\/em> continuous. This is the topology generated by the collection of sets of the form gi<\/sub><\/em>\u20131<\/sup>(V<\/em>), where i<\/em> ranges over I<\/em> and V<\/em> ranges over the open subsets of Yi<\/sub><\/em>. (Note that this collection of sets is itself a set, not a proper class, because they are all subsets of a common set, namely E<\/em>.)<\/p>\n\n\n\n

We model this property categorically as follows. Given any set E<\/em>, the fiber<\/em> above E<\/em> (relative to the functor U<\/em>) is the collection of topological spaces X<\/em> such that U<\/em>(X<\/em>)=E<\/em>. Any such topological space X<\/em> is merely E<\/em> itself, with some topology; and conversely, any topology on E<\/em> gives you a topological space X<\/em> in the fiber above E<\/em>. Hence, instead of talking about topologies on E<\/em>, we can (and will) talk about objects in the fiber above E<\/em>. That is the same thing, but the latter only involves the functor U<\/em>, not any externally defined notion such as a topology.<\/p>\n\n\n\n

What does it mean to find a topology on E<\/em> that would make every gi<\/sub><\/em> continuous? First, formally, saying that gi<\/sub><\/em> is a map from E<\/em> to Yi<\/sub><\/em> is not quite right, categorically, since E<\/em> and Yi<\/sub><\/em> live in different categories. Really, gi<\/sub><\/em> is a map from E<\/em> to U<\/em>(Yi<\/sub><\/em>). Finding a topology on E<\/em> that would make every gi<\/sub><\/em> continuous then means finding an object X<\/em> of Top<\/strong> in the fiber above E<\/em>, first, and one such that gi<\/sub><\/em> lifts<\/em> to a morphism \u011di<\/sub><\/em><\/em><\/em> from X<\/em> to Yi<\/sub><\/em>, for each i<\/em> in I<\/em>. Formally, a lift<\/em> of gi<\/sub><\/em> (with respect to U<\/em>) is a morphism \u011di<\/sub><\/em><\/em><\/em> : X<\/em> \u2192 Yi<\/sub><\/em> in Top<\/strong> whose image by U<\/em> is gi<\/sub><\/em>. Since U<\/em> is faithful, this lift is unique. Informally, that lift is simply the map gi<\/sub><\/em> itself, except that we know that it is continuous from X<\/em> to Yi<\/sub><\/em>.<\/p>\n\n\n\n

A family of morphisms gi<\/sub><\/em> : E<\/em> \u2192 U<\/em>(Yi<\/sub><\/em>), i<\/em> \u2208 I<\/em>, with common domain E<\/em>, is called a U-source<\/em>. We say that a family of morphisms \u011di<\/sub><\/em><\/em>: X<\/em> \u2192 Yi<\/sub><\/em>, i<\/em> \u2208 I<\/em>, is a lift<\/em> of that U<\/em>-source if and if only X<\/em> lies in the fiber above E<\/em> and each \u011di<\/sub><\/em><\/em> is a lift of the corresponding gi<\/sub><\/em>: namely, U<\/em>(X<\/em>)=E<\/em> and U<\/em>(\u011di<\/sub><\/em><\/em><\/em>)=gi<\/sub><\/em> for every i <\/em>in I<\/em>. (The condition U<\/em>(X<\/em>)=E<\/em>, namely that X<\/em> is in the fiber above E<\/em>, is redundant if I<\/em> is non-empty.)<\/p>\n\n\n\n

Good, but I have not just said that you could equip E<\/em> with a topology that makes every gi<\/sub><\/em> continuous. If I had just said that, that would not be much: just equip E<\/em> with the discrete topology. No, I have said that you can give E<\/em> the coarsest<\/em> topology that makes every gi<\/sub><\/em> continuous. One may think of modeling this categorically by requiring that the fiber above E<\/em> be a complete lattice (topologies on a given set indeed form a complete lattice under inclusion, although a rather bizarre one\u2014I may talk about it another day), but there is a sleeker and more comprehensive way.<\/p>\n\n\n\n

Think of it this way. Given the coarsest topology on E<\/em> that makes every gi<\/sub><\/em> continuous (and writing X<\/em> for the topological space obtained by equipping E<\/em> with that topology), how would you show that any given map h<\/em> : Z<\/em> \u2192 X<\/em> from any given topological space Z<\/em> is continuous? I claim that it is enough to show that gi<\/sub><\/em> o h<\/em> : Z<\/em> \u2192 Yi<\/sub><\/em> is continuous for every i<\/em> in I<\/em>. (Pause for a minute, and try to show it by yourself! If gi<\/sub><\/em> o h<\/em> is continuous, then certainly the inverse image of gi<\/sub><\/em>\u20131<\/sup>(V<\/em>) by h<\/em> is open, for any i<\/em> in I<\/em> and for any open subset V<\/em> of Yi<\/sub><\/em>, right?)<\/p>\n\n\n\n

Categorically, this means that for every object Z<\/em> in Top<\/strong>, for every map h<\/em> : U<\/em>(Z<\/em>) \u2192 E<\/em> (in Set<\/strong>), h<\/em> lifts to a morphism \u0125<\/em> from Z<\/em> to X<\/em> (in Top<\/strong>) if and only if every morphism gi<\/sub><\/em> o h<\/em> : U<\/em>(Z<\/em>) \u2192 U<\/em><\/em>(Yi<\/sub><\/em><\/em>) (in Set<\/strong>) lifts to a morphism from Z<\/em> to Yi<\/sub><\/em><\/em> (in Top<\/strong>). In that case, \u011di<\/sub><\/em><\/em> o \u0125<\/em> must be the lift of gi<\/sub><\/em> o h<\/em>, since lifts are unique, owing to the fact that U<\/em> is faithful, and we say that the family of morphisms \u011di<\/sub><\/em><\/em>: X<\/em> \u2192 Yi<\/sub><\/em>, i<\/em> \u2208 I<\/em>, is a U-initial<\/em> lift.<\/p>\n\n\n\n

Property 3: amnesticity (now we are speaking Greek)<\/h3>\n\n\n\n

There is a final, useful property that U<\/em> has, and which is much simpler than its name: amnesticity<\/em>. That is categorical wording for the fact that, given two topologies on the same set E<\/em>, if one is both coarser and finer than the other one, then they are equal; in other words, that the inclusion preordering on the topologies on E<\/em> is an ordering.<\/p>\n\n\n\n

Given any faithful functor U<\/em> from a category C<\/strong> to a category D<\/strong>, every object E<\/em> in D<\/strong> has a fiber, which is the collection of objects X<\/em> in C<\/strong> such that U<\/em>(X<\/em>)=E<\/em>. (That collection may be a proper class.) This is preordered by:<\/p>\n\n\n\n

\n

X \u2264 <\/em>Y (in the fiber above E; read “X is finer than Y”) if and only if the identity morphism on E lifts to a morphism from X to Y in C<\/strong>.<\/p>\n<\/blockquote>\n\n\n\n

In other words, and momentarily returning to the case C<\/strong>=Top<\/strong> and D<\/strong>=Set<\/strong>, X<\/em> and Y<\/em> stand for topologies on E<\/em>, and X<\/em> \u2264 Y<\/em> if and only if the identity map is continuous from X<\/em> to Y<\/em>, if and only if X<\/em> stands for a finer<\/em> topology than the topology that Y<\/em> stands for.<\/p>\n\n\n\n

The definition of topological functors, and basic properties<\/h2>\n\n\n\n

We obtain our definition:<\/p>\n\n\n\n

Definition.<\/strong> A functor U<\/em> : C<\/strong> \u2192 D<\/strong> is topological<\/em> if and only if it is:<\/p>\n\n\n\n