nLab
internal category

Idea

The notion of a category can be formulated internal to any other category with enough pullbacks. By regarding groups as one-object (delooping) groupoids, this generalizes the familiar way in which, for instance

• topological groups are groups internal to topological spaces

• Lie groups are groups internal to smooth manifolds.

An ordinary small category is a category internal to Set.

There is a more general notion of an internal category in a monoidal category, where the pullbacks are replaced by cotensor products.

Definition

Let $A$ be any category. A category internal to $A$ consists of

• an object of objects $C_0\in A$;

• an object of morphisms $C_1\in A$;

together with

• source and target morphisms $s,t:C_1\to C_0$;

• an identity-assigning morphism $e:C_0\to C_1$;

• a composition morphism $c:C_1{\times }_{C_0}{C}_1\to C_1$;

such that the following diagrams commute, expressing the usual category laws:

• laws specifying the source and target of identity morphisms:

$$\begin{array}{ccc}{C}_0& \stackrel{e}{\to }& C_1\\ & 1\searrow & \downarrow s\\ & & C_0\end{array}\begin{array}{ccc}{C}_0& \stackrel{e}{\to }& C_1\\ & 1\searrow & \downarrow t\\ & & C_0\end{array}$$\array{ C_0 & \stackrel{e}{\to} & C_1 \\ {} & 1\searrow & \darr s \\ {} & {} & C_0 } \quad\quad\quad\quad \array{ C_0 & \stackrel{e}{\to} & C_1 \\ {} & 1\searrow & \darr t \\ {} & {} & C_0 }

• laws specifying the source and target of composite morphisms:

$$\begin{array}{ccc}{C}_1{\times }_{C_0}{C}_1& \stackrel{c}{\to }& C_1\\ {}^{p_1}\downarrow & & {\downarrow }^s\\ C_1& \stackrel{s}{\to }& C_0\end{array}\begin{array}{ccc}{C}_1{\times }_{C_0}{C}_1& \stackrel{c}{\to }& C_1\\ {}^{p_2}\downarrow & & {\downarrow }^t\\ C_1& \stackrel{t}{\to }& C_0\end{array}$$\array{ C_1 \times_{C_0} C_1 & \stackrel{c}{\to} & C_1 \\ {}^{p_1}\downarrow & {} & \downarrow^{s} \\ C_1 & \stackrel{s}{\to} & C_0 } \quad\quad\quad\quad \array{ C_1 \times_{C_0} C_1 & \stackrel{c}{\to} & C_1 \\ {}^{p_2}\downarrow & {} & \downarrow^{t} \\ C_1 & \stackrel{t}{\to} & C_0 }

• the associative law for composition of morphisms:

$$\begin{array}{ccc}{C}_1{\times }_{C_0}{C}_1{\times }_{C_0}{C}_1& \stackrel{c{\times }_{C_0}1}{\to }& C_1{\times }_{C_0}{C}_1\\ {}^{1{\times }_{C_0}c}\downarrow & & {\downarrow }^c\\ C_1{\times }_{C_0}{C}_1& \stackrel{c}{\to }& C_1\end{array}$$\array{ C_1 \times_{C_0} C_1 \times_{C_0} C_1 & \stackrel{c\times_{C_0} 1}{\to} & C_1 \times_{C_0} C_1 \\ {}^{1\times_{C_0}c}\downarrow & {} & \downarrow^{c} \\ C_1 \times_{C_0} C_1 & \stackrel{c}{\to} & C_1 }

• the left and right unit laws for composition of morphisms:

$$\begin{array}{ccccc}{C}_0{\times }_{C_0}{C}_1& \stackrel{e{\times }_{C_0}1}{\to }& C_1{\times }_{C_0}{C}_1& \stackrel{1{\times }_{C_0}e}{\leftarrow }& C_1{\times }_{C_0}{C}_0\\ & {}^{p_2}\searrow & {\downarrow }^c& {\swarrow }^{p_1}& \\ & & C_1& & \end{array}$$\array{ C_0 \times_{C_0} C_1 & \stackrel{e \times_{C_0} 1}{\to} & C_1 \times_{C_0} C_1 & \stackrel{1 \times_{C_0} e}{\leftarrow} & C_1 \times_{C_0} C_0 \\ {} & {}^{p_2}\searrow & \downarrow^{c} & \swarrow^{p_1} & {} \\ {} & {} & C_1 & {} & {} }

Here, the pullback $C_1{\times }_{C_0}{C}_1$ is defined via the square

$$\begin{array}{ccc}{C}_1{\times }_{C_0}{C}_1& \stackrel{p_2}{\to }& C_1\\ {}^{p_1}\downarrow & & {\downarrow }^s\\ C_1& \stackrel{t}{\to }& C_0\end{array}$$\array{ C_1 \times_{C_0} C_1 & \stackrel{p_2}{\to} & C_1 \\ {}^{p_1}\downarrow & {} & \downarrow^{s} \\ C_1 & \stackrel{t}{\to} & C_0 }

Notice that inherent to this definition is the assumption that the pullbacks involved actually exist. This holds automatically when the ambient category $A$ has finite limits, but there are some important examples such as $A=$ Diff where this is not the case. Here it is helpful to assume simply that $s$ and $t$ have all pullbacks; in the case of $\mathrm{Diff}$ this occurs if they are submersions.

A groupoid internal to $A$ is all of the above

• such that the cartesian product $C_1\times C_1$ exists

• with a morphism

  $$C_1\stackrel{i}{\to }{C}_1$$C_1 \stackrel{i}{\to} C_1

• such that

  $$\begin{array}{ccc}{C}_1& \stackrel{\mathrm{diag}}{\to }& C_1\times C_1\\ {\downarrow }^s& & {\downarrow }^{c\circ (\mathrm{Id}\times i)}\\ C_0& \stackrel{e}{\to }& C_1\end{array}$$\array{ C_1 &\stackrel{diag}{\to}& C_1 \times C_1 \\ \downarrow^s && \;\;\;\;\;\;\; \downarrow^{c \circ (Id \times i)} \\ C_0 &\stackrel{e}{\to}& C_1 }

• and

  $$\begin{array}{ccc}{C}_1& \stackrel{\mathrm{diag}}{\to }& C_1\times C_1\\ {\downarrow }^t& & {\downarrow }^{c\circ (i\times \mathrm{Id})}\\ C_0& \stackrel{e}{\to }& C_1\end{array}$$\array{ C_1 &\stackrel{diag}{\to}& C_1 \times C_1 \\ \downarrow^t && \;\;\;\;\;\;\; \downarrow^{c \circ (i \times Id)} \\ C_0 &\stackrel{e}{\to}& C_1 }

Alternative definition

If $A$ has all pullbacks, then we can form the bicategory $\mathrm{Span}(A)$ of spans in $A$. A category in $A$ is precisely a monad in $\mathrm{Span}(A)$. The underlying 1-cell is given by the span $(s,t):C_0\leftarrow C_1\to C_0$, and the pullback $C_1{\times }_{C_0}{C}_1$ is the vertex of the composite span $(s,t)\circ (s,t)$. The morphisms $e$ and $c$ are required to be morphisms of spans, which is equivalent to imposing the source and target axioms above. Finally the unit and associativity axioms for monads imply those above.

This approach makes it easy to define the notion of internal profunctor.

Examples

A small category is a category internal to Set. In this case, $C_0$ is a set of objects and $C_1$ is a set of morphisms and the pullback is a subset of the Cartesian product.

Historically, the motivating example was (apparently) the notion of Lie groupoids: a small Lie groupoid is a groupoid internal to the category Diff of smooth manifolds. This generalises immediately to a smooth category?. Similarly, a topological groupoid is a groupoid internal to Top. (Warning: the term ‘topological category’ usually means a topological concrete category, an unrelated notion. Sometimes (e.g. in Jacob Luries ”Higher Topos Theory?”) a ‘topological category’ is defined to be a $\mathrm{Top}$-enriched category, which is a special case of the internal definition if it is interpreted strictly and the collection of objects is small.) In these examples, $C_0$ is a “space of objects” and $C_1$ a “space of morphisms”.

Further examples:

• A cocategory in $C$ is a category internal to $C^{\mathrm{op}}$.
• A double category is a category internal to Cat.
• A double bicategory is a category internal to Bicat (in a suitably weak sense).
• A crossed module is equivalent to a category internal to Grp.
• A Baez-Crans 2-vector space is a category internal to Vect.

Internal functors

Functors between internal categories are defined in a similar fashion. See functor. But if the ambient category does not satisfy the axiom of choice it is often better to use anafunctors instead; this makes sense when $C$ is a superextensive site.

Internal nerve

The idea of the nerve of a small category can be generalised to give an internal nerve construction. Recall (from nerve), the basic idea is that, for a small category, $D$, its nerve, $N(D)$, is a simplicial set whose set of $n$-simplices is the set of sequences of composable morphisms of length $n$ in $D$. This set can be given by a (multiple) pullback of copies of $D_1$. That description will carry across to give a nerve construction for an internal category.

If $C$ is an internal category in some category $A$, (which thus has, at least, the pullbacks required for the constructions to make sense),its nerve $N(C)$ (or if more precision is needed $N_{\mathrm{int}}(C)$, or similar) is the simplicial object in $A$ with

• $N(C{)}_0=C_0$, the ‘object of objects’ of $C$;
• $N(C{)}_1=C_1$, the ‘object of arrows’ of $C$;
• $N(C{)}_2=C_1{\times }_{C_0}{C}_1$ the object of composable pairs of arrows of $C$;
• $N(C{)}_3=C_1{\times }_{C_0}{C}_1{\times }_{C_0}{C}_1$, the object of composable triples of arrows;

and so on. Face and degeneracy morphisms are induced from the structural moprhisms of $C$ in a fairly obvious way.

Internal functors between internal categories induce simplicial morphisms between the corresponding nerves.

Example

A 2-group is an internal category in Grp and so has an internal nerve, which is a simplicial object in Grp, that is a simplicial group. If the 2-group corresponds to a crossed module,$(C\stackrel{\delta }{\to }P)$, then the simplicial group nerve of $(C\stackrel{\delta }{\to }P)$ has Moore complex having $P$ in dimension 0, and $C$ in dimension 1, with the trivial group in all other dimensions. The only possible non-trivial boundary map from dimension 1 to dimension 0 is then the boundary $\delta$ of the crossed module.

Higher internal category

One can also look at this in higher category theory and consider internal n-categories. See

• internal infinity-groupoid

• internal (infinity,1)-category

Related concepts

• weak equivalence of internal categories
• internal site
• internal logic
• enriched category
• locally internal category

References

A survey with an eye towards Lie groupoids is in

• Jean Pradines, In Ehresmann’s footsteps: from Group Geometries to Groupoid Geometries, arXiv:0711.1608

Discussion in terms of monads in spans is in

• Renato Betti, Formal theory of internal categories, Le Matematiche Vol. LI (1996) Supplemento 35-52 pdf

A detailed discussion with emphasis on the correct anafunctor morphisms between internal categories is in

• David Roberts, Internal categories, anafunctors and localization (2010) pdf

Discussion with emphasis on topos theory is in section B.2.3 of

• Peter Johnstone, Sketches of an Elephant

and in section V.7 of

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

An introduction is also in

• John Baez, Alissa Crans, Higher-Dimensional Algebra VI: Lie 2-Algebras

An old discussion on variants of internal categories, crossed modules and 2-groups is archived here.