## You are here

Homeinternal category

## Primary tabs

# internal category

Recall that a small category is a category where the class of objects is a set. As a result, the class of morphisms is also a set. One can thus define a small category completely within set theory, as a 6-tuple $(O,M,s,t,i,c)$, where

1. $O$ is the set of objects and $M$ is the set of morphisms

2. 3. $i:O\to M$ is a function such that $i(A)$ is the identity morphism $1_{A}$

4. $c:K\to M$ is a function such that $c(g,f)$ is the composition of morphism $f$ followed by morphism $g$ (or $g\circ f$); here, $K$ is the collection the all composable pairs of morphisms:

$K=\{(g,f)\in M\times M\mid s(g)=t(f)\}$

These functions satisfy the following rules:

1. the source and target of an identity morphism on an object $A\in O$ is just $A$:

$s(i(A))=t(i(A))=A$ 2. the source of $c(g,f)$ is the the source of $f$, and the target of $c(g,f)$ is the target of $g$:

$s(c(g,f))=s(f)\qquad\mbox{ and }\qquad t(c(g,f))=t(g)$ 3. the composition of a morphism $f$ with the identity morphism of its source $s(f)$ is just $f$; same holds for $t(f)$:

$c(f,i(s(f)))=f=c(i(t(f)),f)$ 4. composition is associative, if defined: that is, if $(g,f),(h,g)\in K$, then

$c(h,c(g,f))=c(c(h,g),f)$

An internal category is the “categorical abstraction” (and generalization) of a small category. Whereas a small category can be completely described in Set, the category of sets, an internal category is completely specified within another category, using only objects and morphisms of this category and their properties.

Definition. Given a category $\mathcal{C}$ with pullbacks, an *internal category* (or *category object*) $\mathcal{D}$ of $\mathcal{C}$ consists of the following:

1. two objects $O,M$ of $\mathcal{C}$, where $O$ is called the

*object of objects*, and $M$ the*object of morphisms*,2. two morphisms $s,t:M\to O$, where $s,t$ are called the

*source*and*target*respectively,3. a morphism $i:O\to M$ called the

*identity*,4. a morphism $c:M\times_{O}M\to M$ called the

*composition*, where $M\times_{O}M$ is the pullback of $s$ and $t$:$\xymatrix@+=2cm{M\times_{O}M\ar[r]^{-}{p_{1}}\ar[d]_{{p_{2}}}&M\ar[d]^{s}\\ M\ar[r]_{t}&O}$

such that the following conditions are satisfied

1. $s\circ i=t\circ i=1_{O}$, the identity morphism on $O$

2. $s\circ c=s\circ p_{2}$ and $t\circ c=t\circ p_{1}$

For condition 3, we need to introduce some notations. By condition 1, we see that $s\circ i\circ t=1_{O}\circ t=t=t\circ 1_{M}$ and $t\circ i\circ s=1_{O}\circ s=s=s\circ 1_{M}$. So we get two commutative diagrams

$\xymatrix@+=2cm{M\ar[r]^{{i\circ t}}\ar[d]_{{1_{M}}}&M\ar[d]^{s}="1"&&M\ar[r]^% {{1_{M}}}\ar[d]_{{i\circ s}}="2"&M\ar[d]^{s}\\ M\ar[r]_{t}&O&&M\ar[r]_{t}&O\ar@{}"1";"2"|-{\mbox{and}}}$ |

Because $M\times_{O}M$ is the pullback of $s$ and $t$, we get two unique morphisms

${i\!\circ\!t\choose 1_{M}}:M\to M\times_{O}M\qquad\mbox{\qquad}{1_{M}\choose i% \!\circ\!s}:M\to M\times_{O}M$ |

and commutative diagrams

$\xymatrix@+=2cm{&M\ar[dl]_{{1_{M}}}\ar[dr]^{{i\circ t}}="1"\ar[d]^{{i\circ t% \choose 1_{M}}}&&&M\ar[dl]_{{i\circ s}}="2"\ar[dr]^{{1_{M}}}\ar[d]^{{1_{M}% \choose i\circ s}}&\\ M&M\times_{O}M\ar[l]^{-}{p_{1}}\ar[r]_{-}{p_{2}}&M&M&M\times_{O}M\ar[l]^{-}{p_% {1}}\ar[r]_{-}{p_{2}}&M\ar@{}"1";"2"|-{\mbox{and}}}$ |

Now, we are ready for condition 3:

- 3.
$c\circ\displaystyle{i\!\circ\!t\choose 1_{M}}=1_{M}=c\circ\displaystyle{i_{M}% \choose i\!\circ\!s}$

Condition 4 also requires some preliminary explanation. Since $\mathcal{C}$ has pullbacks, we get two pullback diagrams:

$\xymatrix@+=2cm{M\times_{O}(M\times_{O}M)\ar[r]^{-}{t\times_{O}1_{M}}\ar[d]&M% \times_{O}M\ar[d]^{{s\circ p_{1}}}&(M\times_{O}M)\times_{O}M\ar[r]\ar[d]_{-}{1% _{M}\times_{O}s}&M\ar[d]^{s}\\ M\ar[r]_{t}&O&M\times_{O}M\ar[r]_{{t\circ p_{2}}}&O}$ |

which result in two morphisms:

$t\times_{O}1_{M}:M\times_{O}(M\times_{O}M)\to M\times_{O}M\qquad\mbox{and}% \qquad 1_{M}\times_{O}s:(M\times_{O}M)\times_{O}M\to M\times_{O}M$ |

Since $M\times_{O}(M\times_{O}M)\cong(M\times_{O}M)\times_{O}M\cong M\times_{O}M% \times_{O}M$, we may view $M\times_{O}M\times_{O}M$ as the domain of morphisms $t\times_{O}1_{M}$ and $1_{M}\times_{O}s$. We are now ready for condition 4:

- 4.
$c\circ(t\times_{O}1_{M})=c\circ(1_{M}\times_{O}s)$.

Remark. In Set, an internal category is just a small category as we have seen from the discussion earlier. An internal category in Cat is a double category.

## Mathematics Subject Classification

18D35*no label found*18D99

*no label found*18D05

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections