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I write a mathematical research monograph.

It has a section named "Partially ordered categories". The first subsection (with the most basic properties of partially ordered categories) of this section has the same name ("Partially ordered categories").

Is it OK for a section and its subsection to share the same name?

If not, what should I rename the subsection to?

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The short answer to your general question is "no." If the subsection is titled the same as the section, then either one of them is named incorrectly or else your outline is incorrect. An outline is supposed to be like this:

Title: Great Cities of the World

I. American Cities

A. Large Cities

  1. New York, NY

  2. Chicago, IL

B. Historical Cities

  1. Philadelphia, PA

  2. San Diego, CA

II. European Cities

etc. etc. etc.

In mathematical terms: Every subsection must be a limited subset of the section in which it occurs.

Corollary One: A section must not be divided into subsections unless there are at least two subsections.

Corollary Two: A section name is implicitly part of all subsection names beneath it.

Corollary Three: If explicitly adding the section name to a subsection name would cause the subsection name to not make sense, then the subsection is in the wrong section.

To answer your specific question: The subsection should be named "Basic Properties."

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4

You definitely want to avoid the kind of clash you're describing.

Intuitively, the section is composed of smaller subsections, which expand on particular areas. So a clash feels like you're saying "And now, within the topic of (Topic X), let's talk about (Topic X)." Even if there's some kind of sense to it, it's confusing and should be avoided.

The key to solving this is figuring out what the difference is between the larger section, and the subsection. If they're both talking about (Topic X) - in this case, about Partially Ordered Categories - then the subsection must be discussing a specific aspect of (Topic X) - otherwise, you wouldn't have any other subsections in the same category.

Most often, if you feel like the names should be the same, it's because you're discussing a large topic, and at some point you want to formally define the topic. For example:

  • Topic X
    • Introduction
    • Motivation for Topic X
    • Topic X
    • Use #1 of Topic X
    • Use #2 of Topic X
    • Variations on Topic X
    • Topic X in Popular Culture

In this case, the role of the subsection might be:

  • A formal definition of the topic
  • Basic facts about the topic
  • A brief overview of the topic, which will be expanded upon in later subsections

...or other similar roles. You've got to figure out the particular role your subsection plays, in order to give it an accurate name.

I'd suggest that the easiest way to do this is to try to make the title of the clashing subsection longer. If this subsection is talking about Topic X, but isn't everything about Topic X, then explaining (in as many words as it takes) what the subsection is about will give you a more specific title.

For example, in your particular case, the relevant subsection contains only the formal mathematical definition. So if you'd describe the subsections in your own words, you'd say "This subsection has the definition of Partially ordered categories, and then the next subsection introduces another concept that works with this definition." So you could use:

3.5 Partially ordered categories

3.5.1 Definition

3.5.2 Dagger categories

Or even

3.5 Partially ordered categories

3.5.1 Definition of Partially ordered categories

3.5.2 Dagger categories

and you'll be entirely clear.

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