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Thinking through value propositions

Business types use the jargon “value proposition” for the (typically specific and unique) values offered by a product or service. When I started Groupprops four years ago, and incorporated it into Subject Wikis 2.5 years ago, I had some ideas about the value proposition, but these ideas have been continually modified based on the way people have actually used the website.

As I shift attention to improving Topospaces significantly to incorporate algebraic topology as well as the relationships between point set, algebraic, and basic differential topology, I think it’s a good time to reflect on the experience of building Groupprops and think about whether the experience can help with a more rapid development of Topospaces.

Information at fingertips

One of the things that I disliked about most conventional reference resources was the fact that specific information details were often left for readers to work out, and the information that was given was scattered across multiple sources. I’ve found this extremely frustrating. It may be a good approach to teaching a course, but it’s not helpful in a reference. One of the goals of subject wikis is to make specific information quickly and readily accessible to people in a way that they can eyeball it and get a sense of things.

This was not completely missing from the original goals, but my original focus had been more to include proofs of general statements. Now, information on each specific example in full, gory detail has taken on a lot of significance. Also, unlike the case of most references, these examples or particular cases/instances are treated as separate entities with their full development, rather than mentioned only in the context of general results that they may or may not illustrate. Some additional observations about the significance of this:

  • In textbooks, examples are usually developed when the context is ripe, and are developed only to the extent that they illustrate some important principle. Thus, a lot of examples that don’t illustrate any principle directly important to the author of the textbook are ignored. The subject wikis approach is different. Each example is developed as a separate entity. Then, for a particular example, general facts that highlight the particular features of that example are linked to (and necessary computations to elucidate the link are shown). Conversely, for a general fact, those examples that are best understood in context of that fact are linked to. Neither specific examples nor general facts are parasitic on each other.

  • In many areas of learning, there is a bunch of common misconceptions that students develop. Part of the source of these misconceptions is the fact that students haven’t seen enough examples of a sufficiently wide range. In the subject wikis, the goal is to design pages in a manner that scrolling through the page gives a “general feel” for the nature of examples, thus reducing the chance of misconceptions. For commonly identified misconceptions, cautionary notes and highlights are included both in the general page and in pages developing each specific example.

  • Separately developing examples without worrying about whether those examples are “important” makes the resource useful for discovery — somebody can come along and see a well-developed example and notice patterns that perhaps weren’t otherwise obvious.

For instance, consider the page element structure of symmetric group:S4. In addition to developing information about the element structure of this particular group, the page explains how this element structure fits into the context of interpretations of the group as a symmetric group, as well as a projective general linear group of degree two. Links are provided to general facts about the element structure of symmetric groups and also of projective general linear groups of degree two.

Relationships between facts, and identifying the critical jumps

As should be clear from what’s said so far, our goal is to not shy from the gory details of computation and development of specific examples. Even more important here are the explicit calculations that underlie big theorems, results, and relationships, often calculations that are left as an exercise to the reader in too many references. In addition to providing these calculations, the context, generalization, and limitations of these calculations are considered. This could help people get a feel of exactly why and how the calculations work as they do.

For instance, those who’ve seen some point set topology and the beginnings of algebraic topology may have encountered the notion of fundamental group. In a typical algebraic topology text, the definition is accompanied by a brief description of why the fundamental group is a group. The details are often omitted or left as an exercise for the reader. The Topospaces page on fundamental group takes a different approach. Each of the aspects of showing that the fundamental group is a group is stated clearly, and the proof/explanation is deferred to a separate page, where the construction is covered in detail (these pages are still under development). It turns out that many of the same ideas turn up when we are trying to understand loop spaces, so this same proof/explanation page serves double duty as showing things about loop spaces.

Thinking deeply about simple things

Textbooks are often written for first-time learners or perhaps second-time learners. They use a linear ordering and, typically, have to restrict information on a topic to what can be explained based on topics covered so far.

It is often the case that the definitions of the simple and basic concepts in a subject (particularly in mathematics) have a number of subtleties that cannot be pointed out to first-time learners when the concepts are being introduced. These subtleties are mentioned in random places as people learn. Ten years after learning a subject, the experienced student has no single place to turn to to get an updated, improved, but concise description of all these added nuances.

With the subject wikis, we hope to overcome this limitation. Definitions of simple ideas are accompanied by alternative definitions/interpretations that rely on the development of more complicated machinery. The equivalence of these multiple definitions may itself rely on important results of the subject, that are linked to. Experts in a subject usually have all these multiple definitions in their head when they think of the concept, whereas novices are often stuck on the “basic” definition and can recall other definitions only on being prompted (even if they are aware of these). By having these multiple definitions all available in one single place, people can build their expertise in the topic faster. The pages on 2-subnormal subgroup and normal subgroup at Groupprops are illustrations.