internal category



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

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.


Internal category

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

  • an object of objects C 0AC_0 \in A;

  • an object of morphisms C 1AC_1 \in A;

together with

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

  • laws specifying the source and target of identity morphisms:
C 0 e C 1 1 s C 0C 0 e C 1 1 t C 0 \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:
C 1× C 0C 1 c C 1 p 1 s C 1 s C 0C 1× C 0C 1 c C 1 p 2 t C 1 t C 0 \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:
C 1× C 0C 1× C 0C 1 c× C 01 C 1× C 0C 1 1× C 0c c C 1× C 0C 1 c C 1 \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:
C 0× C 0C 1 e× C 01 C 1× C 0C 1 1× C 0e C 1× C 0C 0 p 2 c p 1 C 1 \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× C 0C 1C_1 \times_{C_0} C_1 is defined via the square

C 1× C 0C 1 p 2 C 1 p 1 s C 1 t C 0 \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 AA has finite limits, but there are some important examples such as A=A =\, Diff where this is not the case. Here it is helpful to assume simply that ss and tt have all pullbacks; in the case of DiffDiff this occurs if they are submersions.

Internal groupoid

A groupoid internal to AA is all of the above

  • with a morphism

    C 1iC 1 C_1 \stackrel{i}{\to} C_1
  • such that

    t=(C 1iC 1sC 0),s=(C 1iC 1tC 0). t = ( C_1 \stackrel{i}{\to} C_1 \stackrel{s}{\to} C_0 ),\;\;\;\; s = ( C_1 \stackrel{i}{\to} C_1 \stackrel{t}{\to} C_0 ).
  • and

    C 1 diag C 1 t× C 0 tC 1 Id×i C 1 t× C 0 sC 1 s c C 0 e C 1 \array{ C_1 &\stackrel{diag}{\to}& C_1\;{}_t \times_{C_0}{}_t C_1 &\stackrel{Id \times i}{\to}& C_1\;{}_t \times_{C_0}{}_s C_1 \\ \downarrow^s &&&& \; \downarrow^c \\ C_0 &&\stackrel{e}{\to}&& C_1 }
  • and

    C 1 diag C 1 s× C 0 sC 1 i×Id C 1 t× C 0 sC 1 t c C 0 e C 1 \array{ C_1 &\stackrel{diag}{\to}& C_1\;{}_s \times_{C_0}{}_s C_1 &\stackrel{i \times Id}{\to}& C_1\;{}_t \times_{C_0}{}_s C_1 \\ \downarrow^t &&&& \; \downarrow^c \\ C_0 &&\stackrel{e}{\to}&& C_1 }

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 CC is a superextensive site.

Alternative definition

If AA has all pullbacks, then we can form the bicategory Span(A)Span(A) of spans in AA. A category in AA is precisely a monad in Span(A)Span(A). The underlying 1-cell is given by the span (s,t):C 0C 1C 0(s,t) : C_0 \leftarrow C_1 \to C_0, and the pullback C 1× C 0C 1C_1 \times_{C_0} C_1 is the vertex of the composite span (s,t)(s,t)(s,t) \circ (s,t). The morphisms ee and cc 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.

Internal nerve

The notion of nerve of a small category can be generalised to give an internal nerve construction. For a small category, DD, its nerve, N(D)N(D), is a simplicial set whose set of nn-simplices is the set of sequences of composable morphisms of length nn in DD. This set can be given by a (multiple) pullback of copies of D 1D_1. That description will carry across to give a nerve construction for an internal category.

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

  • N(C) 0=C 0N(C)_0 = C_0, the ‘object of objects’ of CC;
  • N(C) 1=C 1N(C)_1 = C_1, the ‘object of arrows’ of CC;
  • N(C) 2=C 1× C 0C 1N(C)_2 = C_1 \times_{C_0} C_1 the object of composable pairs of arrows of CC;
  • N(C) 3=C 1× C 0C 1× C 0C 1N(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 CC in a fairly obvious way.

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

Higher internal categories

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


A small category is a category internal to Set. In this case, C 0C_0 is a set of objects and C 1C_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 a ‘topological category’ is defined to be a TopTop-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 0C_0 is a “space of objects” and C 1C_1 a “space of morphisms”.

Further examples:


In a cartesian closed category

If the ambient (finitely complete) category E\mathbf{E} is a cartesian closed category, then the category Cat(E)Cat(\mathbf{E}) of categories internal to E\mathbf{E} is also cartesian closed. This was proved twice by Charles and Andrée (under her maiden name Bastiani) Ehresmann using generalised sketches, or may be proven directly as follows (see also Johnstone, remark after B2.3.15):


Let E\mathbf{E} be a finitely complete cartesian closed category. Then the category Cat(E)Cat(\mathbf{E}) of internal categories in EE is also finitely complete and cartesian closed.


First suppose E\mathbf{E} is finitely complete. Then the category of directed graphs E \mathbf{E}^{\bullet \stackrel{\to}{\to} \bullet} is also finitely complete, and since Cat(E)Cat(\mathbf{E}) is monadic over E \mathbf{E}^{\bullet \stackrel{\to}{\to} \bullet}, it follows that Cat(E)Cat(\mathbf{E}) is also finitely complete.

Now suppose that E\mathbf{E} is finitely complete and cartesian closed. Let Δ 3\Delta_3 denote the category of nonempty ordinals up to and including the ordinal with 4 elements. We have a full and faithful embedding

N:Cat(E)E Δ 3 opN \colon Cat(\mathbf{E}) \to \mathbf{E}^{\Delta_3^{op}}

where the codomain category is cartesian closed. Indeed, the exponential of two objects FF, GG in E Δ 3 op\mathbf{E}^{\Delta_3^{op}} may be computed as an E\mathbf{E}-enriched end

G F(m)= n f:nmG(n) F(n)G^F(m) = \int_n \prod_{f \colon n \to m} G(n)^{F(n)}

when evaluated at mOb(E Δ 3 op)m \in Ob(\mathbf{E}^{\Delta_3^{op}}), as is easily checked (see for instance here); note that this end is a finite limit diagram since Δ 3\Delta_3 is finite.

If CC, DD are internal categories in E\mathbf{E}, seen as functors Δ 3 opE\Delta_3^{op} \to \mathbf{E}, the exponential NC NDN C^{N D} defines the exponential in Cat(E)Cat(\mathbf{E}). To see this, it suffices to check that NC NDN C^{N D}, as defined by the end formula above, is a category BB, i.e., is in the essential image of the nerve functor. For in that case, we have natural isomorphisms

F×DCinCat(E)NF×NDN(F×D)NCinE Δ 3 opNFNC NDinE Δ 3 opFBinCat(E)\frac{ \frac{F \times D \to C \;\;\;\text{in}\; Cat(\mathbf{E})} {N F \times N D \cong N(F \times D) \to N C \;\;\;\text{in}\; \mathbf{E}^{\Delta_3^{op}}}} {\frac{N F \to N C^{N D} \;\;\;\text{in}\; \mathbf{E}^{\Delta_3^{op}}} {F \to B \;\;\;\text{in}\; Cat(\mathbf{E})}}

whence BB satisfies the universal property required of an exponential.

Objects in the essential image of the nerve NN are characterized as functors Δ 3 opE\Delta_3^{op} \to \mathbf{E} which take intervalic joins in Δ 3\Delta_3 to pullbacks in E\mathbf{E}, as given precisely by the Segal conditions. The remainder of the proof is then finished by the following lemma.


If C:Δ 3 opEC \colon \Delta_3^{op} \to \mathbf{E} satisfies the Segal conditions and X:Δ 3 opEX \colon \Delta_3^{op} \to \mathbf{E} is any functor, then C XC^X also satisfies the Segal conditions.


For any XX we have the formula

C X(m)= kHom(Xk, f:nk g:nmC(n)).C^X(m) = \int_k Hom(X k, \prod_{f \colon n \to k} \prod_{g \colon n \to m} C(n)).

Since the enriched end and the internal hom-functor Hom(Xk,)Hom(X k, -) both preserve pullbacks, we are reduced to checking that

  • If CC satisfies the Segal conditions, then so does
    f:nk g:nmC(n)\prod_{f \colon n \to k} \prod_{g \colon n \to m} C(n)

    as a functor Δ 3 opE\Delta_3^{op} \to \mathbf{E} in the argument mm (for each fixed kk).

Note that the displayed statement is a proposition in the language of finitely complete categories (i.e., in finitary essentially algebraic logic). Since hom-functors E(e,):ESet\mathbf{E}(e, -) \colon \mathbf{E} \to Set jointly preserve and reflect the validity of such propositions, it suffices to prove it for the case where E=Set\mathbf{E} = Set. But this is classical elementary category theory; it says precisely that if CC is a small (ordinary) category, then the usual functor categories C 2C^{\mathbf{2}}, C 3C^{\mathbf{3}} are equivalently described by exponentials of (truncated) simplicial sets. This completes the proof.

In a topos

If the ambient category is a topos, then with the right kind of notion of internal functor, the internal groupoids form the corresponding (2,1)-topos of groupoid-valued stacks and the internal categories form the corresponding 2-topos of category-valued stacks/2-sheaves.

For the precise statement see at 2-topos – In terms of internal categories


A survey with an eye towards Lie groupoids is in

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

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

and in section V.7 of

An introduction is also in

The original proofs that the category of internal categories is cartesian closed when the ambient category is finitely complete and cartesian closed are in

and in

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

Revised on May 1, 2016 10:52:23 by Volodymyr Lyubashenko (