1. Foundational assumptions [000R]

[001F] Definition 1·a.

A meta-category $\mathfrak{C}$ is defined by explaining what an object of $\mathfrak{E}$ is, and, given two objects $x,y\in \mathfrak{E}$, what a morphism from $x$ to $y$ is, together with the following operations:

  1. for each object $x\in \mathfrak{E}$, an identity map $\Idn{x} : x \to x$,
  2. for any two maps $f:x\to y$ and $g:y\to z$, a composite map $f;g : x \to z$,
  3. such that the following equations hold: \[ \Idn{x};h = h\qquad h;\Idn{y} = h\qquad f;(g;h) = (f;g);{h} \]

In our definition above, we have not imposed any restrictions on what kinds of things the objects and morphisms are; our definition is pre-mathematical, so we do not assume beforehand that there is a such thing as a collection of “all” meta-categories.

Remark. We may define analogous notions of meta-functor, etc. But we do not assume that the notion of “all meta-functors $\mathfrak{C}\to\mathfrak{D}$” is well-defined; the notion is entirely schematic.

Assumption. We assume a meta-category $\BoldSymbol{\mathfrak{Coll}}$ whose objects we will refer to as “collections”. We assume that the meta-category of all collections satisfies the axioms of Lawvere’s ETCS.

[001G] Definition 1·b.

A category $E$ is defined to be a meta-category whose objects are defined to be the elements of some collection, and for any two objects $x,y\in E$ the morphisms $x\to y$ are defined to be the elements of some collection.

Consequently there exists a meta-category $\BoldSymbol{\mathfrak{Cat}}$ of all categories. Following Lawvere (but deviating from some other authors that ground the notion of meta-categories in classes) we notice that $\BoldSymbol{\mathfrak{Cat}}$ is cartesian closed; in other words, all functor categories exist regardless of size.

Assumption. At times we may assume that there exists a category $\SET\subseteq\BoldSymbol{\mathfrak{Coll}}$ of collections that we will refer to as sets, such that $\SET$ is closed under the axioms of ETCS. Rather than work with $\SET$ at all times, our approach is to use the tools of relative category theory to objectify the notions of “small” and “locally small” category over any category $B$, generalizing the role of $\SET$ from classical category theory.