Introduction

This wiki is written in the modern, homotopy-coherent dialect. The intention is to keep statements short by not repeating the qualifier “(∞,1)” on every other word. The terminology below is fixed once for the whole site, and individual entries should not redefine it.

If you arrive from a 1-categorical background, just read every “category” as “(∞,1)-category” and add the strictness adjective when you want it.

Body

Categorical conventions

  • category = 1-category = (∞,1)-category. All categories on this wiki are $(\infty,1)$-categories by default. Every notion (limits, colimits, adjunctions, presheaves, …) is the $(\infty,1)$-categorical one.

  • $n$-category = $(\infty, n)$-category. The integer counts the depth of non-invertible morphisms; everything above level $n$ is invertible. So a $2$-category on this wiki means an $(\infty,2)$-category, not a strict 2-category.

  • classical category is the explicit name for the 1-categorical ($\mathsf{Set}$-enriched) notion. Whenever the 1-categorical sense is needed — to contrast with the homotopy-coherent one, or to discuss classical theorems — the qualifier classical is mandatory.

  • anima denotes the $(\infty,1)$-category of $\infty$-groupoids (a.k.a. spaces, homotopy types, $\mathsf{An}$). Reserved word.

A consequence of these conventions: phrases like “the category of presentable ∞-categories” become “the category of presentable categories”, and “a 1-category in the classical sense” becomes “a classical category”.

Suggested entry structure

Every entry should have three blocks. The first two are written by hand; the third is rendered automatically by the layout.

  1. Introduction — motivation and intuition. What is the entry about, and why does it deserve a name? What does it generalise or specialise? What goes wrong without it?
  2. Body — formal definitions, characterisations, the main statements. Subheadings encouraged when there is more than one route in.
  3. Neighborhood — auto-rendered. Lists outgoing $\text{wikilinks}$, incoming references (backlinks), and a 1-hop force-directed graph centred on the current entry.

Entries that need a long bibliography should add a Pointers section just before the auto-rendered Neighborhood.

Linking

Use double-bracket wikilinks for cross-references:

A [[Compactly generated category]] is  [[Presentable category]] whose generators are $\aleph_0$-compact.

The bracketed text is matched against entry titles and aliases: in the frontmatter, case-insensitive. Unresolved links render as a greyed dashed underline so they show up while drafting.