{"id":6613,"date":"2023-04-20T21:34:18","date_gmt":"2023-04-20T19:34:18","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6613"},"modified":"2023-05-15T07:42:42","modified_gmt":"2023-05-15T05:42:42","slug":"topological-functors-ii-the-cartesian-closed-category-of-c-maps","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6613","title":{"rendered":"Topological Functors II: the Cartesian-closed category of C-maps"},"content":{"rendered":"\n

A while back<\/a>, I gave an introduction to topological functors, and I promised I would post one or more followups. So here we are. I will explain how the construction of the Cartesian-closed category of C<\/strong>-generated topological spaces due to M. Escard\u00f3, J. Lawson and A. Simpson [3] is really a construction that generalizes way beyond topological spaces, to all (well-fibered) topological constructs [2, Section 3]. I applied it to streams and prestreams in [2], but not everyone may be interested in the latter, while knowing that the idea of C<\/strong>-generation adapts to general topological functors is probably more interesting.<\/p>\n\n\n\n

I will definitely not explain the whole construction. It turns out that this would be too much. I will concentrate on the core of the construction, which generalizes the first step of Escard\u00f3, Lawson and Simpson: the construction of a Cartesian-closed category Map<\/strong>C<\/sub><\/span>, built from a so-called strongly productive class C<\/span> of objects of a category C<\/strong>, where C<\/strong> is the domain of a so-called well-fibered topological construct U<\/em>. The Escard\u00f3-Lawson-Simpson construction is the special case where C<\/strong> is the category Top<\/strong> of topological spaces and U<\/em> is the forgetful functor to Set<\/strong>.<\/p>\n\n\n\n

(Update, May 15th, 2023: I have prepared a video on this subject, to be presented at the seminar on categorical topology at the University of Puebla, Mexico, see this page<\/a>. The slides and the video(s) are at the bottom. This is probably better for an introduction to the topics on this page. But continue reading if you wish to understand the technical details, too.)<\/p>\n\n\n\n

A refresher on topological functors<\/h2>\n\n\n\n

What you should keep in mind is that the paradigmatic example of a topological functor is the forgetful functor from the category Top<\/strong> of topological spaces to the category Set<\/strong> of sets. (I will call this the standard example<\/em>.) The actual definition is more complicated: a functor U<\/em> : C<\/strong> \u2192 D<\/strong> is topological<\/em> if and only if it is faithful, amnestic, and is such that every U<\/em>-source has a U<\/em>-initial lift.<\/p>\n\n\n\n

Before I start, you may safely assume that D<\/strong>=Set<\/strong>. In any case, the construction I want to explain here eventually only works in that case. A topological functor U<\/em> from a category C<\/strong> to Set<\/strong> is called a topological construct<\/em>.<\/p>\n\n\n\n

Alright. Being faithful<\/em> is easy to explain. That just means that U<\/em> is injective on hom sets, namely that if two morphisms f<\/em>, g<\/em> : X<\/em> \u2192 Y<\/em> in C<\/strong> (with the same<\/em> domain and the same<\/em> codomain) are such that U<\/em>(f<\/em>)=U<\/em>(g<\/em>), then f<\/em>=g<\/em>. In the standard example, any two continuous maps between the same spaces that are equal as ordinary functions between the underlying sets… are equal.<\/p>\n\n\n\n

Given two objects X<\/em> and Y<\/em> of C<\/strong>, any morphism g<\/em> from U<\/em>(X<\/em>) to U<\/em>(Y<\/em>) in D<\/strong> either lifts<\/em> to a morphism \u011d<\/em> from X<\/em> to Y<\/em> in C<\/strong>, meaning that U<\/em>(\u011d<\/em>)=g<\/em>, or it does not. Since U<\/em> is faithful, if the lift \u011d<\/em> exists, it is unique. In the standard example, a set function g<\/em> between two topological spaces has a lift if and only if it is continuous, and then \u011d<\/em> is g<\/em> itself, seen as a continuous map. Hence, and to avoid some pedantry, I will simply write the lift \u011d<\/em>… as g<\/em>. I will in fact avoid talking about lifts altogether, and I will simply say that g<\/em> is<\/em> a C<\/strong>-morphism from X<\/em> to Y<\/em>, instead of saying that g<\/em> lifts to one. (And I am tempted to even say “C<\/strong>-continuous”, but maybe it is enough if you think of C<\/strong> as being some form of stand-in for C<\/strong>ontinuous.)<\/p>\n\n\n\n

The condition “every U-source has a U-initial lift<\/em>” means the following. A U<\/em>-source is 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>. (Note that each C<\/strong>-object Yi<\/sub><\/em> is part of the datum, not just U<\/em>(Yi<\/sub><\/em>). In the standard example, this means we are given topological spaces<\/em> Yi<\/sub><\/em>, not just their underlying sets.) A lift<\/em> of that U<\/em>-source is a C<\/strong>-object X<\/em> such that U<\/em>(X<\/em>)=E<\/em> (in the standard example, X<\/em> must be E<\/em> with some topology) that turns every gi<\/sub><\/em> into a C<\/strong>-morphism from X<\/em> to Yi<\/sub><\/em>.<\/p>\n\n\n\n

This lift is U<\/em>-initial if and only if, in a sense, X<\/em> is the coarsest C<\/strong>-object such that U<\/em>(X<\/em>)=E<\/em> and making every gi<\/sub><\/em> into a C<\/strong>-morphism. More formally, U<\/em>-initiality means that for every C<\/strong>-object Z<\/em>, an arbitrary D<\/strong>-morphism h<\/em> : U<\/em>(Z<\/em>) \u2192 E<\/em> is in fact a C<\/strong>-morphism from Z<\/em> to X<\/em> if and only if gi<\/sub><\/em> o h<\/em> is a C<\/strong>-morphism from Z<\/em> to Yi<\/sub><\/em> for every i<\/em> in I<\/em>.<\/p>\n\n\n\n

Speaking of “C<\/strong>-objects X<\/em> such that U<\/em>(X<\/em>)=E<\/em>” is a bit awkward. It is traditional to talk about those C<\/strong>-objects as those being in the fiber<\/em> of E<\/em>. In the standard example, the fiber of a set E<\/em> consists of those topological spaces whose points are those of E<\/em>. Therefore, the fiber of E<\/em> is really another way of talking about the collection of topologies on E<\/em> (in the standard example).<\/p>\n\n\n\n

Topologies are ordered by “coarser than” and “finer than”, and the categorical analogue is to say that given two C<\/strong>-objects X<\/em> and Y<\/em> in the fiber of the same D<\/strong>-object E<\/em>, X<\/em> is finer than Y<\/em> (in notation, X<\/em>\u2264Y<\/em>) if and only if the identity morphism from E<\/em> to E<\/em> is a C<\/strong>-morphism from X<\/em> to Y<\/em>. Naturally, Y<\/em> is coarser than X<\/em> if and only if X<\/em> is finer than Y<\/em>.<\/p>\n\n\n\n

Then U<\/em> is amnestic<\/em> if and only if \u2264 is antisymmetric on each fiber. In the standard example, this means that if a topology is both coarser and finer than another, then they are equal.<\/p>\n\n\n\n

In the final paragraphs of part I<\/a>, I had given a few examples of topological functors, including the forgetful functor from the category Pre<\/strong> of preordered sets to Set<\/strong>, as well as functors from categories of streams or prestreams to Top<\/strong>.<\/p>\n\n\n\n

We will need a final property that not all topological functors have. A functor U<\/em> : C<\/strong> \u2192 D<\/strong> is well-fibered<\/em> if and only if:<\/p>\n\n\n\n