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Internalization

Table of Contents

1. Overview

The intuition is, just as we define a "mathematical gadget" to consist of "stuff" equipped with "structure" such that a bunch of "properties" hold, we can model this by using objects instead of "stuff", morphisms for "structure", and commutative diagrams for "properties". This gives us an "internal gadget" relative to a category.

2. Example: Internal Category

Let \(A\) be a category with pullbacks. An Category Object internal to \(A\) consists of

  • an object of objects \(C_{0}\in A\)
  • an object of morphisms \(C_{1}\in A\)

equipped with

  • source and target morphisms \(s,t\colon C_{1}\to C_{0}\)
  • an identity-assigning morphism \(e\colon C_{0}\to C_{1}\)
  • a composition morphism \(c\colon C_{1}\times_{C_{0}}C_{1}\to C_{1}\)

such that the following diagrams expressing category laws hold, specifying the source of identity morphisms:

\begin{equation} \require{AMScd} \begin{CD} C_{0} @>e>> C_{1}\\ @V1VV @VVsV \\ C_{0} @= C_{0} \end{CD} \end{equation}

For the target of identity morphisms

\begin{equation} \begin{CD} C_{0} @>e>> C_{1}\\ @V1VV @VVtV \\ C_{0} @= C_{0} \end{CD} \end{equation}

The source of a composite morphism:

\begin{equation} \begin{CD} C_{1}\times_{C_{0}}C_{1} @>c>> C_{1}\\ @Vp_{1}VV @VVsV \\ C_{1} @>s>> C_{0} \end{CD} \end{equation}

The target of a composite morphism:

\begin{equation} \begin{CD} C_{1}\times_{C_{0}}C_{1} @>c>> C_{1}\\ @Vp_{2}VV @VVsV \\ C_{1} @>t>> C_{0} \end{CD} \end{equation}

Associativity for composition of morphisms:

\begin{equation} \begin{CD} C_{1}\times_{C_{0}}C_{1}\times_{C_{0}}C_{1} @>c\times_{C_{0}}1>> C_{1}\times_{C_{0}}C_{1}\\ @V1\times_{C_{0}}cVV @VVcV \\ C_{1}\times_{C_{0}}C_{1} @>c>> C_{0} \end{CD} \end{equation}

The left and right unit laws for composition:

\begin{equation} \begin{CD} C_{1}\times_{C_{0}}C_{1} @>e\times_{C_{0}}1>> C_{1}\times_{C_{0}}C_{1} @<1\times_{C_{0}}e<< C_{1}\times_{C_{0}}C_{1}\\ @Vp_{2}VV @VVcV @VVp_{2}V \\ C_{1} @= C_{1} @= C_{1} \end{CD} \end{equation}

Note, here the pullback \(C_{1}\times_{C_{0}}C_{1}\) is defined by the square

\begin{equation} \begin{CD} C_{1}\times_{C_{0}}C_{1} @>p_{2}>> C_{1}\\ @Vp_{1}VV @VVsV \\ C_{1} @>t>> C_{0} \end{CD} \end{equation}

2.1. Internal Functors

When the ambient category \(A\) does not satisfy the axiom of choice, it is often better to use anafunctors instead of "vanilla functors".

Let \(C\), \(D\) be internal categories in some ambient category \(A\), then an Internal Functor \(F\colon C\to D\) is

  • a morphism of objects \(F_{0}\colon C_{0}\to D_{0}\) in \(A\)
  • a morphism of morphisms \(F_{1}\colon C_{1}\to D_{1}\) in \(A\)

such that it respects the source map

\begin{equation} \begin{CD} C_{1} @>f_{1}>> D_{1}\\ @VsVV @VVsV \\ C_{0} @>f_{0}>> D_{0} \end{CD} \end{equation}

Functors respect the target map:

\begin{equation} \begin{CD} C_{1} @>f_{1}>> D_{1}\\ @VtVV @VVtV \\ C_{0} @>f_{0}>> D_{0} \end{CD} \end{equation}

It respects identity morphisms:

\begin{equation} \begin{CD} C_{0} @>f_{0}>> D_{0}\\ @VeVV @VVeV \\ C_{1} @>f_{1}>> D_{1} \end{CD} \end{equation}

Functors preserve composing "internal morphisms":

\begin{equation} \begin{CD} C_{1}\times_{C_{0}}C_{1} @>f_{1}\times_{C_{0}}f_{1}>> D_{1}\times_{D_{0}}D_{1}\\ @VcVV @VVcV \\ C_{1} @>f_{1}>> D_{1} \end{CD} \end{equation}

where \(f_{1}\times_{C_{0}}f_{1}\) is the unique morphisms defined by the universal property of pullbacks.

3. Other Examples

4. References

  • John C. Baez, Alissa S. Crans,
    "Higher-Dimensional Algebra VI: Lie 2-Algebras".
    arXiv:math/0307263, 50 pages. §2 defines internal categories, internal functors, internal natural transformations
  • Bryce Clarke,
    "Internal split opfibrations and cofunctors".
    arXiv:2004.00187, 25 pages. Definitions 2.1 and 2.2 for internal categories and internal functors
  • nLab entries:
  • Saunders MacLane, Ieke Moerdijk,
    Sheaves in Geometry and Logic.
    Section V.5–7 (V.5 is about internal vs external, V.6 for internal groups, V.7 for internal categories)
  • Saunders MacLane,
    Categories for the Working Mathematician.
    Springer, Second ed., ch XII, section 1.
  • David Roberts,
    "Internal categories, anafunctors and localisations".
    arXiv:1101.2363, def 4.7 for internal natural transformations.

Last Updated 2023-09-05 Tue 08:37.