GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 6, 1997, 533-556

INTERNAL CATEGORIES IN A LEFT EXACT COSIMPLICIAL CATEGORY

D. PATARAIA

Abstract. The notion of an internal category in a left exact cosimplicial category is introduced. For any topos over sets a certain left exact cosimplicial category is constructed functorially and the category of internal categories in it is investigated. The notion of a fundamental group is defined for toposes admitting the notion of “a discrete category.”

Introduction

Our primary interest in this paper is to introduce the notion of an internal category in a left exact cosimplicial category, generalizing ordinary internal categories. We call a cosimplicial category left exact when all of its components are categories with finite limits and all coface and codegeneracy functors preserve them.

In Section 1 the definitions are given of an internal category in a category (which is wellknown) as well as in a left exact cosimplicial category.

In Section 2 for any topos over Sets a certain left exact cosimplicial category is constructed functorially and the category of internal categories in it is investigated. Two examples of these constructions are considered. These examples correspond to the case where the topos is the category of sheaves over a locally compact topological space, and where it is a topos of presheaves.

In Section 3 we consider toposes which admit the notion of “a discrete internal category.” For such toposes we determine the notion of fundamental group by means of the “discrete” category corresponding to the terminal object. When the topos is the category of sheaves over a locally compact and locally simply connected space, then its fundamental group is the same as the fundamental group of the underlying space.

Some words about the notation:

533 Sets denotes the category of sets. Cat denotes the 2-category of categories, functors, and natural transfor-

1991 Mathematics Subject Classification. 76E15.

Key words and phrases. Topos, fundamental group, cosimplicial category.

1072-947X/97/1100-0533$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

mations.

The 2-category of toposes, geometric morphisms, and natural transformations will be denoted by 𝔗𝔬𝔭. Let Top be the full sub-2-category of 𝔗𝔬𝔭, consisting of all toposes over Sets, i.e., objects of Top are such toposes, from whic there exists a geometric morphism in Sets (note that if such a morphism exists, it is unique up to a natural isomorphism). The product in the category Top will be denoted by ×_{Set}.

Internal categories and internal functors in the category of topological spaces will be called continuous categories and continuous functors, respectively.

I denotes the topological space of real numbers between 0 and 1, I = [0; 1].

Internal categories in Cat will be called double categories [1].

For any category C with finite limits and any finite diagram f in it, we will denote by pr_{x}, or simply by pr, the natural projection from the limit of the diagram f (i.e., from \underleftarrow{lim}(f)) to the term x (provided that this causes no ambiguity). For example, if f is of the shape x ⟶ y ⟵ z, then pr_{x} is the projection: pr_{x} : \underleftarrow{lim}(x ⟶ y ⟵ z) ⟶ x. For any product x × y × z × ⋯ in C we denote by pr_{i} the projection of this product to the i + 1-th term.

Let Cat(C) be the category of internal categories in C (we mean C with finite limits). Consider the forgetful functor from Cat(C) to C, which sends each internal category in C to its object of objects. We denote the right adjoint functor of this forgetful functor by ad(−). Then for any object x ∈ C, the object of objects of the internal category ad(x) will be x, the object of morphisms x × x, and morphisms of source and target will be respectively the projections pr_{0}, pr_{1} from x × x to x. (Note that ad(x) is the antidiscrete internal category over x.)

1. The Notion of an Internal Category in a Left Exact Cosimplicial Category

First we introduce some useful notions.

The category Δ has as objects all finite ordinal numbers [n] = 0, 1, …, n, and as arrows f : [n] ⟶ [m] all (weakly) monotone functions.

In the category Δ we choose arrows:

δ_{n}^{i} : [n − 1] ⟶ [n] for 0 ≤ i ≤ n is the injective monotone function whose image omits i;

σ_{n}^{i} : [n + 1] ⟶ [n] for 0 ≤ i ≤ n is the surjective monotone function having the same value on i and i + 1;

ρ_{n}^{i,j} : [1] ⟶ [n] for 0 ≤ i ≤ j ≤ n is the function: ρ_{n}^{i,j}(0) = i, ρ_{n}^{i,j}(1) = j; ρ_{n}^{i} : [0] ⟶ [n] for 0 ≤ i ≤ n is the function for which ρ_{n}^{i}(0) = i.

Sometimes we will write these arrows omitting the subscript n.

As is wellknown, a cosimplicial (resp., simplicial) object in the category C is determined as a covariant (resp., contravariant) functor from Δ to C. For any cosimplicial object F, we will write as usual F_{n} (resp., δ_{n}^{i}, σ_{n}^{i}, ρ_{n}^{i,j}, ρ_{n}^{i}) instead of F [n] (resp., instead of F (δ_{n}^{i}), F (σ_{n}^{i}), F (ρ_{n}^{i,j}), F (ρ_{n}^{i})).

We call a cosimplicial object in the category of categories Cat a cosimplicial category.

Definition 1.

(1) A cosimplicial category F is called left exact if for any n≥0, F_{n} is a category with finite limits and all functors δ^{i} : F_{n−1} ⟶ F_{n}, σ^{i} : F_{n+1} ⟶ F_{n} preserve them.

(2) A natural transformation α between two left exact cosimplicial categories F and F′ is called left exact if for any n≥0 the functor α : F_{n} ⟶ F′_{n} preserves all finite limits.

Remark 2. It is easy to check that any simplicial topos determines a left exact cosimplicial category by taking inverse image functors.

Now we will give the definitions of an internal category in a category and in a left exact cosimplicial category. The first definition is the wellknown one, but we need both definitions to compare them.

Definition 3. Let C be a category with finite limits. By an internal category X in C we mean a sixtuple (X_{1}, X_{0}, ∂_{1}, ∂_{0}, s, m), where X_{1} and X_{0} are the objects of C (called the object of objects and the object of morphisms, respectively);

∂_{1} : X_{1} ⟶ X_{0}, ∂_{0} : X_{1} ⟶ X_{0}, s : X_{0} ⟶ X_{1},

m : \underleftarrow{lim}(X_{1} ⟶^{∂_{0}} X_{0} ⟵^{∂_{1}} X_{1}) ⟶ X_{1}

are morphisms in C (called morphisms of domain, codomain, identity, and composition, respectively), such that the following diagrams are commutative:

[figure: X_{0}; X_{0} ,; X_{1}; X_{1}; id_{X_{0}}; s; ∂_{1}; s; ∂_{0}]  (1)

[figure: X_{1}; X_{1}; \underleftarrow{lim}(X_{1} ⟶^{∂_{1}} X_{0} ⟵^{∂_{0}} X_{1}); X_{0}; pr_{1}; m; ∂_{1}; ∂_{1}]  (2)

(here, pr_{1} is the projection of \underleftarrow{lim}(X_{1} ⟶ X_{0} ⟵ X_{1}) to the left-hand X_{1}).

[figure: X_{1}; X_{1}; \underleftarrow{lim}(X_{1} ⟶^{∂_{1}} X_{0} ⟵^{∂_{0}} X_{1}); X_{0}; m; pr_{2}; ∂_{0}; ∂_{0}]  (3)

(here pr_{2} is the projection of \underleftarrow{lim}(X_{1} ⟶ X_{0} ⟵ X_{1}) to the right-hand X_{1}).

\underleftarrow{lim} (\begin{matrix}X_{1} & & & & X_{1} & & & & X_{1} \\ & ↘^{∂_{0}} & & ^{∂_{1}}↙ & & ↘^{∂_{0}} & & ^{∂_{1}}↙ \\ & & X_{0} & & & & X_{0} \end{matrix})

[figure]
\underleftarrow{lim} (\begin{matrix}X_{1} & & & & X_{1} \\ & ↘^{∂_{0}} & & ^{∂_{1}}↙ & & ↘^{pr_{1}} \\ & & X_{0} & & & & X_{2} \end{matrix})  \underleftarrow{lim} (\begin{matrix}& & X_{1} & & & & X_{1} \\ & ^{pr_{2}}↙ & & ↘^{∂_{0}} & & ^{∂_{1}}↙ \\ X_{2} & & & & X_{0} \end{matrix})
||by the diagram (2)  ||by the diagram (3)
\underleftarrow{lim} (\begin{matrix}X_{1} & & & & X_{1} \\ & ↘^{∂_{0}} & & ^{∂_{1}}↙ & & ↘^{m} \\ & & X_{0} & & & & X_{2} \end{matrix})  \underleftarrow{lim} (\begin{matrix}& & X_{1} & & & & X_{1} \\ & ^{m}↙ & & ↘^{∂_{0}} & & ^{∂_{1}}↙ \\ X_{2} & & & & X_{0} \end{matrix})
↓^{pr}  ↓^{pr}
\underleftarrow{lim} (\begin{matrix}X_{1} & & & & X_{1} \\ & ↘^{∂_{0}} & & ^{∂_{1}}↙ \\ & & X_{0} \end{matrix})  \underleftarrow{lim} (\begin{matrix}X_{1} & & & & X_{1} \\ & ↘^{∂_{0}} & & ^{∂_{1}}↙ \\ & & X_{0} \end{matrix})

[figure: X_{1}; m; m]

(4)

[figure: X_{1}; X_{1} ,; ( \begin{matrix}s∘∂_{1} id_{X_{1}} \\ ∂_{1} \end{matrix} ); id_{X_{1}}; \underleftarrow{lim} (\begin{matrix}X_{1} & & & & X_{1} \\ & ↘^{∂_{0}} & & ^{∂_{1}}↙ \\ & & X_{0} \end{matrix}); m] (5)

[figure: X_{1}; X_{1} .; ( \begin{matrix}id_{X_{1}} s∘∂_{0} \\ ∂_{0} \end{matrix} ); id_{X_{1}}; \underleftarrow{lim} (\begin{matrix}X_{1} & & & & X_{1} \\ & ↘^{∂_{0}} & & ^{∂_{1}}↙ \\ & & X_{0} \end{matrix}); m]  (6)

Definition 4. Let F be a left exact cosimplicial category. By an internal category X in F we mean a sixtuple (X_{1}, X_{0}, ∂_{1}, ∂_{0}, s, m)

where X_{1} is the object of F_{0} (called the object of objects), X_{0} is the object of F_{1} (called the object of morphisms), ∂_{1} : X_{1} ⟶ δ^{1}(X_{0}), ∂_{0} : X_{1} ⟶ δ^{0}(X_{0}) are the morphisms in F_{1} (called

the morphisms of domain and codomain, respectively), s : X_{0} ⟶ σ(X_{1}) is the morphism of F_{0} (called the identity morphism), m : \underleftarrow{lim}(ρ_{2}^{0,1}(X_{1}) ⟶ ρ_{2}^{1}(X_{0}) ⟵ ρ_{2}^{1,2}(X_{1})) ⟶ ρ_{2}^{0,1}(X_{1}) is the morphism in

F_{2} (called the morphism of composition), such that the following diagrams are commutative:

[figure: X_{0}; X_{0} ,; X_{1}; X_{1}; id_{X_{0}}; s; s; σ ∘ δ^{1}(X_{0}); σ ∘ δ^{0}(X_{0}); σ(∂_{1}); σ(∂_{0})]  (1′)

[figure: X_{2}; ρ_{2}^{0,1}(X_{1}); ρ_{2}^{0,2}(X_{1}); ρ_{2}^{0,1}(δ^{1}(X_{0})); ρ_{2}^{0,2}(δ_{1}(X_{0})) ,; ρ_{2}^{0,1}(∂_{1}); ρ_{2}^{0,2}(∂_{1}); pr; m; ρ_{2}^{0}(X_{0})]  (2′)

[figure: X_{2}; ρ_{2}^{0,2}(X_{1}); ρ_{2}^{1,2}(X_{1}); ρ_{2}^{0,2}(δ^{0}(X_{0})); ρ_{2}^{1,2}(δ_{0}(X_{0})) ,; ρ_{2}^{0,2}(∂_{0}); ρ_{2}^{1,2}(∂_{0}); m; pr; ρ_{2}^{0}(X_{0})]  (3′)

\underleftarrow{lim} ( ↘^{ρ{0,1 \atop 3}(∂_{0})} ↙^{ρ{1,2 \atop 3}(∂_{1})} ↘^{ρ{1,2 \atop 3}(∂_{0})} ↙ ρ{2,3 \atop 3}(∂_{1})_{ρ_{3}^{1}(X_{0})  ρ_{3}^{2}(X_{0})}^{ρ_{3}^{0,1}(X_{1})  ρ_{3}^{1,2}(X_{1})  ρ_{3}^{2,3}(X_{1})} )

[figure]
\underleftarrow{lim} ( ↘^{ρ{0,1 \atop 3}(∂_{0})} ↙ ρ{1,2 \atop 3}(∂_{1}) ↖^{δ^{0}(pr)}_{ρ_{3}^{1}(X_{0})=(δ^{0}∘ρ_{2}^{0})(X_{0}) δ^{0}(X_{2})}^{ρ_{3}^{0,1}(X_{1}) ρ_{3}^{1,2}(X_{1})=(δ^{0}∘ρ_{2}^{0,1})(X_{1})} ) \underleftarrow{lim} ( ↗^{δ^{3}(pr)} ↘^{ρ{1,2 \atop 3}(∂_{0})} ↙ ρ{2,3 \atop 3}(∂_{1})_{δ^{0}(X_{2}) (δ^{3}∘ρ_{2}^{2})(X_{0})=ρ_{3}^{2}(X_{0})}^{(δ^{3}∘ρ_{2}^{1,3})(X_{1})=ρ_{3}^{1,3}(X_{1}) ρ_{3}^{2,3}(X_{1})} )
||by diagram (2′)  ||by diagram (3′)
\underleftarrow{lim} ( ↘^{ρ{0,1 \atop 3}(∂_{0})} ↙ (δ^{0}∘ρ{0,2 \atop 2})(∂_{1}) ↖^{δ^{0}(m)}_{ρ_{3}^{1}(X_{0})=(δ^{0}∘ρ_{2}^{0})(X_{0}) δ^{0}(X_{2})}^{ρ_{3}^{0,1}(X_{1})  (δ^{0}∘ρ_{2}^{0,1})(X_{1})} ) \underleftarrow{lim} ( ↗^{δ^{3}(m)} ↘^{(δ^{3}∘ρ{0,2 \atop 2})(∂_{0})} ↙ ρ{2,3 \atop 3}(∂_{1})_{δ^{0}(X_{2}) (δ^{3}∘ρ_{2}^{2})(X_{0})=ρ_{3}^{2}(X_{0})}^{(δ^{3}∘ρ_{2}^{1,3})(X_{1}) ρ_{3}^{2,3}(X_{1})} )
↓^{pr}  ↓^{pr}  (4′)
\underleftarrow{lim} ( ↘^{ρ{0,1 \atop 3}(∂_{0})} ↙ δ^{0}∘ρ{0,2 \atop 2}(∂_{1})_{ρ_{3}^{1}(X_{0})=(δ^{0}∘ρ_{2}^{0})(X_{0})}^{ρ_{3}^{0,1}(X_{1}) (δ^{0}∘ρ_{2}^{0,1})(X_{1})} )  \underleftarrow{lim} ( ↘^{δ^{3}∘ρ{0,2 \atop 2}(∂_{0})} ↙ ρ{2,3 \atop 3}(∂_{1})_{(δ^{3}∘ρ_{2}^{2})(X_{0})=ρ_{3}^{2}(X_{0})}^{(δ^{3}∘ρ_{2}^{1,3})(X_{1}) ρ_{3}^{2,3}(X_{1})} )
||  ||
δ^{2}(X_{2}) δ^{1}(X_{2})

[figure: δ^{2}(m); δ^{1}(m)]

(δ^{2} ∘ δ^{1})(X_{1}) = (δ^{1} ∘ δ^{1})(X_{1}) ,

[figure: X_{1}; (σ^{0} ∘ δ^{1})(X_{1}),; σ^{0}(X_{2}); \underleftarrow{lim} ( ↘^{δ^{1}∘σ(∂_{1})}  ^{∂_{1}}↙_{(δ^{1}∘^{1})(X_{0})=δ^{1}(X_{0})}^{(δ^{1}∘σ)(X_{1})  X_{1}} ); ( \begin{matrix}δ^{1}(s)∘∂_{1} id_{X_{1}} \\ ∂_{1} \end{matrix} ); id_{X_{1}}; σ^{0}(X_{2})] (5′)

[figure: X_{1} .; (σ^{1} ∘ δ^{1})(X_{1}); σ^{1}(X_{2}); \underleftarrow{lim} ( ↘^{δ^{1}∘σ(∂_{1})}  ^{∂_{0}}↙_{(δ^{0}∘^{0})(X_{0})=δ^{0}(X_{0})}^{(δ^{1}∘σ)(X_{1})  X_{1}} ); ( \begin{matrix}δ^{0}(s)∘∂_{0} id_{X_{1}} \\ ∂_{0} \end{matrix} ); id_{X_{1}}; σ^{1}(X_{2})] (6′)

An internal functor between two internal categories X and Y in the left exact cosimplicial category F consists of two morphisms:

f_{0} : X_{0} ⟶ Y_{0} in F_{0} and f_{1} : X_{1} ⟶ Y_{1} in F_{1} commuting with ∂_{1}, ∂_{0}, s and m.

We denote the category of internal categories and functors in F by Cat(F).

Any left exact natural transformation α between two left exact cosimplicial categories F and F′ induces the functor Cat(α) from Cat(F) to Cat(F′);

Cat(α) : Cat(F) ⟶ Cat(F′),

X=(X_{0}, X_{1}, ∂_{1}, ∂_{0}, s, m)↦(α_{0}(X_{0}), α_{1}(X_{1}), α_{1}(∂_{1}), α_{1}(∂_{0}), α_{0}(s), α_{2}(m)).

2. Constructing a Left Exact Cosimplicial Category from a Topos

Definition 5. Let E be a topos over Sets.

(1) A simplicial topos Ẽ : Δ^{op} ⟶ Top is defined as follows:

Ẽ_{0} is the topos E, Ẽ_{1} is the topos E ×_{Sets} E,

Ẽ_{2} is the topos E ×_{Sets} E ×_{Sets} E, ⋯, Ẽ_{n} is the topos \underbrace{E ×_{Sets} ⋯ ×_{Sets} E}_{n+1},

⋯,

for each 0 ≤ i ≤ n

∂_{i} = Ẽ(δ_{n}^{i}) = (pr_{0}, pr_{1}, …, pr_{i−1}, pr_{i+1}, …, pr_{n}) :
: \underbrace{E ×_{Sets} ⋯ ×_{Sets} E}_{n+1} ⟶ \underbrace{E ×_{Sets} ⋯ ×_{Sets} E}_{n} (omitting i)

and

s_{i} = Ẽ(σ_{n}^{i}) = (pr_{0}, pr_{1}, …, pr_{i−1}, pr_{i}, pr_{i}, pr_{i+1}, …, pr_{n}) :
: \underbrace{E ×_{Sets} ⋯ ×_{Sets} E}_{n+1} ⟶ \underbrace{E ×_{Sets} ⋯ ×_{Sets} E}_{n+2} .

One can easily see that ∂_{i} and s_{i} are the geometric morphisms which satisfy the usual axioms of simplicial objects [2].

(2) The left exact cosimplicial category Δ(E) is defined as follows:

for n ≥ 0 let Δ(E)_{n} be the same category as Ẽ_{n} and for each 0 ≤ i ≤ n let δ_{n}^{i} : Δ(E)_{n−1} ⟶ Δ(E)_{n} (resp. σ_{n}^{i} : Δ(E)_{n+1} ⟶ Δ(E)_{n}) be the inverse image of the geometric morphism ∂_{i}^{n} : Ẽ_{n} ⟶ Ẽ_{n−1} (resp. s_{i}^{n} : Ẽ_{n} ⟶ Ẽ_{n+1}).

It readily follows that for any n ≥ 0 the category Δ(E)_{n} has finite limits (because it is a topos) and any coface and codegeneracy functor preserves them (because it is the inverse image functor of some geometric morphism).

The category of internal categories in Δ(E) is denoted by Cat Δ(E).

Definition 6. For any topos E over Sets let S(E) be the category, whose objects are all pairs (P, ϕ), where P is an internal category in Top, ϕ is an internal functor ϕ : P ⟶ ad(E) such that ϕ_{0} : P_{0} ⟶ E, and ϕ_{1} : P_{1} ⟶ E ×_{Sets} E are local homeomorphisms of toposes.

A morphism between the pairs (p, ϕ) and (P′, ϕ′) is an internal functor f : P ⟶ P′, such that ϕ′ ∘ f = ϕ.

Remark 7.

(1) For any E_{1}, E_{2} ∈ Top, the geometric morphism f : E_{1} ⟶ E_{2} with the functor f_{!} ⊣ f^{∗}, where f_{!} preserves equalizers, will be called as usual a local homeomorphism. In that case (see [3]) the category E_{1} is equivalent to the comma category E_{2}/f_{!}(1) (here 1 means the terminal object).

(2) S(−) is a contravariant functor from Top to Cat. Any geometric morphism f : E ⟶ E′ induces a functor S(f) : S(E′) ⟶ S(E) as follows: for (P′, ϕ′) ∈ S(E′), S(f) is a pair (P, ϕ) such that the squares

\begin{matrix}P_{0} & ⟶ & P′_{0} \\ ^{ϕ_{0}}↓ & & ↓^{ϕ′_{0}} \\ E & ⟶_{f} & E′ \end{matrix},  P_{1} ⟶ P′_{1} ^{ϕ′_{1}}↓  ↓^{ϕ′_{1}} E ×_{Set} E ⟶_{f×f} E′ ×_{Set} E′

are pullbacks. By [3, Corollary 4.35] it is clear that since in the above commutative diagrams the geometric morphisms ϕ′_{0} and ϕ′_{1} are the local homeomorphisms, ϕ_{0} and ϕ_{1} will be the local homeomorphisms too.

Theorem 8. Let E be a topos over Sets. Then there is an equivalence of categories

Cat Δ(E) ⇆^{Φ_{E}}_{Ψ_{E}} S(E)

and this equivalence is natural in E.

Proof. First let us construct the functor Φ_{E}. Suppose (X_{0}, X_{1}, ∂_{1}, ∂_{0}, s, m)∈ Cat Δ(E). Then X_{0} ∈ E and X_{1} ∈ E ×_{Set} E. Denote the comma categories

E/X_{0}, E ×_{Set} E/X_{1} by E_{0} and E_{1}, respectively.

For i = 0, 1 the morphisms ∂_{i} : X_{1} ⟶ δ^{i}(X_{0}) in E ×_{Set} E induce by

[3, corollary 4.35] the geometric morphisms ¯∂_{i} : E_{1} ⟶ E_{0} such that the diagrams

\begin{matrix}E_{1} & ⟶^{¯∂_{1}} & E_{0} \\ ↓ & & ↓ \\ E & ⟶^{∂_{1}} & E \end{matrix} , \begin{matrix}E_{1} & ⟶^{¯∂_{0}} & E_{0} \\ ↓ & & ↓ \\ E & ⟶^{∂_{0}} & E \end{matrix}

are commutative.

Similarly, the morphisms s : X_{0} ⟶ σ(X_{1}), m : \underleftarrow{lim}(ρ_{2}^{0,1}(X_{1}) ⟶^{ρ_{2}^{0,1}}

ρ_{2}^{1}(X_{0}) ⟵^{ρ_{2}^{1,2}} ρ_{2}^{1,2}(X_{1})) ⟶ ρ_{2}^{0,2}(X_{1}) induce the geometric morphisms s̄ : E_{0} ⟶ E_{1} and m̄ : \underleftarrow{lim}(E_{1} ⟶ E_{0} ⟵ E_{1}) ⟶ E_{1} for which the diagrams

E_{0} ⟶^{s̄} E_{1} ↓  ↓ E ⟶^{(1,1)} E ×_{Set} E ,  \underleftarrow{lim}(E_{1} → E_{0} ← E_{1}) ⟶^{m̄} E_{1} ↓  ↓ E ×_{Set} E ×_{Set} E  ⟶^{(pr_{0},pr_{1})} E ×_{Set} E

are commutative.

Using (1′)–(6′) for (X_{0}, X_{1}, ∂_{1}, ∂_{0}, s, m) one can easily prove that the sixtuple (E_{0}, E_{1}, ¯∂_{1}, ¯∂_{0}, s̄, m̄) satisfies conditions (1)-(6) for an internal category. Hence Φ_{E}((X_{0}, X_{1}, ∂_{1}, ∂_{0}, s, m)) is determined as a pair ((E_{0}, E_{1}, ¯∂_{1}, ¯∂_{0}, s̄, m̄), (α_{0}, α_{1})) where α_{0} and α_{1} are natural geometric morphisms from the comma category into the underlying topos.

Now let us construct the functor Ψ_{E}. Let ((E_{0}, E_{1}, ¯∂_{1}, ¯∂_{0}, s̄, m̄), (α_{0}, α_{1}))∈ S(E). Then ((E_{0}, E_{1}, ¯∂_{1}, ¯∂_{0}, s̄, m̄) is an internal category in Top and (α_{0}, α_{1}) is an internal functor from ((E_{0}, E_{1}, ¯∂_{1}, ¯∂_{0}, s̄, m̄) to ad(E) such that α_{0} : E_{0} ⟶ E and α_{1} : E_{1} ⟶ E ×_{Set} E are local homeomorphisms. Suppose the

local homeomorphisms α_{0} and α_{1} are induced by X_{0} ∈ E and X_{1} ∈ E ×_{Set} E

respectively. By [3, Corollary 4.35] the morphisms ¯∂_{1}, ¯∂_{0}, s̄, m̄ induce the morphisms ∂_{1} : X_{1} ⟶ δ^{1}(X_{0}),∂_{0} : X_{1} ⟶ δ^{0}(X_{0}), s : X_{0} ⟶ σ(X_{1}),

m : \underleftarrow{lim}(ρ_{2}^{0,1}(X_{1}) ⟶^{ρ_{2}^{0,1}(∂_{0})} ρ_{2}^{1}(X_{0}) ⟵^{ρ_{2}^{1,2}(∂_{0})} ρ_{2}^{1,2}(X_{1})) ⟶ ρ_{2}^{0,2}(X_{1}) (here the

functors δ, σ and ρ are from the cosimplicial category Δ(E)).

The fact that the sixtuple (X_{0}, X_{1}, ∂_{1}, ∂_{0}, s, m) satisfies conditions (1′)– (6′) follows from the fact that (E_{0}, E_{1}, ¯∂_{1}, ¯∂_{0}, s̄, m̄) satisfies conditions (1)– (6) for an internal category. Therefore Ψ_{E}(((E_{0}, E_{1} ¯∂_{1}, ¯∂_{0}, s̄, m̄), (α_{0}, α_{1}))) is determined to be the sixtuple (X_{0}, X_{1}, ∂_{1}, ∂_{0}, s, m).

It is easily seen that the functors Φ_{E}, Ψ_{E} are inverse to each other.

For any E ∈ Top we define the forgetful functor T_{E} : S(E) ⟶ E as follows: for any ((E_{0}, E_{1}, ¯∂_{1}, ¯∂_{0}, s̄, m̄), (α_{0}, α_{1})) ∈ S(E) let T_{E}(((E_{0}, E_{1}, ¯∂_{1}, ¯∂_{0}, s̄, m̄), (α_{0}, α_{1}))) be X ∈ E such that E_{0} ˜=E/X.

If a geometric morphism f : E ⟶ E′ is given, then its inverse image functor f^{∗} is interchangeable with the functor T_{∗}, i.e., the diagram

S(E) ⟵^{S(f)} S(E′)
T_{E}↓  ^{T_{E′}} ↓
E ⟵^{f^{∗}} E′

is commutative.

Definition 9. Let P = ((E_{0}, E_{1}, ∂_{1}, ∂_{0}, s, m), (α_{0}, α_{1})) ∈ S(E).

(1) We define the opposite internal category P^{op} of P to be that obtained by interchanging ∂_{1} and ∂_{0}, and “twisting” the definition of m, i.e., in the internal category P^{op} we must replace (E_{0}, E_{1}, ∂_{1}, ∂_{0}, s, m) by its opposite.

(2) We call P an internal groupoid if the underlying internal category (E_{0}, E_{1}, ∂_{1}, ∂_{0}, s, m) is an internal groupoid in Top.

Remark 10. For any Q = (X_{0}, X_{1}, ∂_{1}, ∂_{0}, s, m) ∈ Cat Δ(E), the opposite category of Q is Q^{op} = (X_{0}, X_{1}^{∗}, ∂_{1}^{∗}, ∂_{0}^{∗}, s, m^{∗}), where X_{1} is (pr_{1}, pr_{0})^{∗}(X_{1}), ∂_{i}^{∗} = (pr_{1}, pr_{0})^{∗}(∂_{i}) for i = 0, 1 and m^{∗} = (pr_{2}, pr_{1}, pr_{0})^{∗}(m); here

(pr_{1}, pr_{0}) : E ×_{Sets} E ⟶ E ×_{Sets} E is the “swapping” geometric morphism.

Note that internal groupoids in Δ(E) form a full subcategory in S(E). Denote this subcategory by Gpd Δ(E) or simply by Gpd(E).

It is easy to see that Gpd(−) is a functor (as S(−)); for any f : E ⟶ E′ the geometric morphism in Top, S(f) : S(E′) ⟶ S(E) assigns the internal groupoid in Gpd(E) to each internal groupoid in Gpd(E′).

We will consider two examples of our constructions.

Example 1. Consider the case with E as a topos of sheaves over a locally compact Hausdorff topological space A. Denote this topos by shv(A). In this case, for each n ≥ 0 \underbrace{shv(A) ×_{Sets} ⋯ ×_{Sets} shv(A)}_{n+1} = shv(A^{n+1}) [3],

any geometric morphism from shv(A^{n+1}) to shv(A^{m+1}) for n, m ≥ 0 is induced by a continuous map from A^{n+1} to A^{m+1} [4], and this geometric morphism is a local homeomorphism if and only if the continuous map between the underlying topological spaces is a local homeomorphism. Therefore the category S(shv(a)) can be represented as a category of pairs (C, α), where C = (C_{0}, C_{1}, ∂_{1}, ∂_{0}, s, m) is a continuous category, α = (α_{0}, α_{1}) : C ⟶ ad(A) is a continuous functor, such that both maps α_{0} : C_{0} ⟶ A and α_{1} : C_{1} ⟶ A×A are local homeomorphisms. Let us denote this new representation of the category S(shv(a)), i.e., the representation by continuous categories and functors, simply by S(A).

Now we will derive some technical properties of some elements of S(A) to be used in Section 3.

Proposition 11. Let A be a topological space as above and let (C, α) ∈

S(A). Then the continuous map C_{1} ⟶^{(∂_{1},∂_{0})} C_{0} × C_{0} satisfies the unique lifting property of paths [5]. This means that for any two paths (ω, ω′) : I ⟶ C_{1} in C_{1}, for which (∂_{1}, ∂_{0}) ∘ ω = (∂_{1}, ∂_{0}) ∘ ω′ and ω(0) = ω′(0), we have ω = ω′ (a path is a continuous function I ⟶ C_{1}).

Proof. Suppose there exist two paths ω, ω′ : I ⟶ C_{1} with (∂_{1}, ∂_{0}) ∘ ω = (∂_{1}, ∂_{0}) ∘ ω′, ω(0) = ω′(0) and ω ≠ ω′. The proposition will be proved in three steps. In the 1st step it will be shown that (∂_{1}, ∂_{0}) : C_{1} ⟶ C_{0} × C_{0} is a local homeomorphism; in the 2nd step a sequence p_{n} will be found, which has two distinct limits in C_{1}, and in the 3rd step a new sequence p̃_{n} will be constructed using the sequence p_{n}, which leads to a contradiction.

1st step. Consider the commutative diagram

[figure: C_{1}; C_{0} × C_{0}; A × A; α_{0} × α_{0} .; α_{1}; (∂_{1}, ∂_{0})]

Since in this diagram α_{1} and α_{0} × α_{0} are local homeomorphisms, (∂_{1}, ∂_{0}) is a local homeomorphism too [5].

2nd step. Denote by κ_{i} for i = 0, 1 the path ∂_{i} ∘ ω′ = ∂_{i} ∘ ω in C_{0}. Consider the path κ = (κ_{1}, κ_{0}) in C_{0} × C_{0} and the object (P ⟶^{∂} I) = κ^{∗}((∂_{1}, ∂_{0}) : C_{1} ⟶ C_{0} × C_{0}) ∈ shv(I). Two paths ω and ω′ in C_{1} induce two global sections of the local homeomorphism ∂ : P ⟶ I or, equivalently, two global elements ω̄, ω̄′ : 1 ⟶ (∂ : P ⟶ I) in shv(I). Let ι : U ↣ 1 be an equalizer of the parallel arrows ω̄ and ω̄′. We can represent the subobject ι : U ↣ I as an open subset U of I. This is a subspace of those t ∈ I for which ω(t) = ω′(t). Therefore 0 ∈ U. Consider a maximal open connected subspace in I which contains 0 ∈ I. Since ω ≠ ω′, this component of the connectivity of U has the shape [0, t_{0}). It is clear that 0 < t_{0} < 1 and ω(t_{0}) ≠ ω′(t_{0}). Denote ω(t_{0}) by p and ω′(t_{0}) by p′. Choose a sequence t_{n} in [0, t_{0}) whose limit is t_{0}. Then for n ≥ 0, ω(t_{n}) = ω′(t_{n}); denote this latter by p_{n}. Thus we have constructed a sequence p_{n} in C_{1} which has two distinct limits p and p′.

3rd step. Consider κ_{1}(t_{0}) ∈ C_{0} (resp., κ_{0}(t_{0}) ∈ C_{0}) and s(κ_{1}(t_{0})) ∈ C_{1} (resp., s(κ_{0}(t_{0})) ∈ C_{1}). Since (∂_{1}, ∂_{0}) is a local homeomorphism, there exists an open neighborhood W_{1} (resp., W_{0}) of s(κ_{1}(t_{0})) (resp., s(κ_{0}(t_{0}))) which is mapped homeomorphically onto the open neighborhood U_{1} × U_{1} (resp., U_{0} × U_{0}) of (κ_{1}(t_{0}), κ_{1}(t_{0})) (resp., of (κ_{0}(t_{0}), κ_{0}(t_{0}))). Denote the inverse of the homeomorphism (∂_{1}, ∂_{0})|_{W_{1}} : W_{1} ⟶ U_{1} × U_{1} (resp., (∂_{1}, ∂_{0})|_{W_{0}} : W_{0} ⟶ U_{0} × U_{0}) by f_{1} (resp., by f_{0}).

Consider an open neighborhood U_{1} × U_{0} of (κ_{1}(t_{0}), κ(t_{0})) in C_{0} × C_{0}. Since lim_{n→∞}(d_{1}, d_{0})(p_{n}) = (κ_{1}(t_{0}), κ_{0}(t_{0})), there exists n_{0} > 0 such that

n > n_{0} implies (∂_{1}, ∂_{0})(p_{n}) ∈ U_{1} × U_{0}. Denote by q_{n}^{1} (resp., q_{n}^{0}) the point (κ_{1}(t_{0}), ∂_{1}(p_{n})) ∈ C_{0} × C_{0} (resp., (∂_{0}(p_{n}), κ_{0}(t_{0})) ∈ C_{0} × C_{0}). Obviously, lim_{n→∞} q_{n}^{1} = (κ_{1}(t_{0}), κ_{1}(t_{0})) (resp., lim_{n→∞} q_{n}^{0} = (κ_{0}(t_{0}), κ_{0}(t_{0})) and

lim_{n→∞} f_{1}(q_{n}^{1}) = sκ_{1}(t_{0}) (resp., lim_{n→∞} f_{0}(q_{n}^{0}) = sκ_{0}(t_{0})).

Now consider the following sequence in C_{1}:

p̃_{n} = f_{0}(q_{n}) ∘ p_{n} ∘ f_{1}(q_{n}^{1}), n > n_{0};

here ∘ means the internal composition in the continuous category C. We have

lim_{n→∞} p̃_{n} = lim_{n→∞} f_{0}(q_{n}^{0}) ∘ lim_{n→∞} p_{n} ∘ lim_{n→∞} f_{1}(q_{n}^{1}) =

= s(κ_{0}(1)) ∘ lim_{n→∞} p_{n} ∘ s(κ_{1}(t_{0})) = lim_{n→∞} p_{n}.

But p and p′ are limits of p_{n} so that the limits of p̃_{n} will be p and p′. For each n > n_{0}, p̃_{n} ∈ (∂_{1}, ∂_{0})^{−1}(κ_{1}(t_{0}), κ_{0}(t_{0})) and the subspace

(∂_{1}, ∂_{0})^{−1}(κ_{1}(t_{0}), κ_{0}(t_{0})) ⊂ C_{1} is discrete, since (∂_{1}, ∂_{0}) is a local homeomorphism. Therefore in the discrete topological space (∂_{1}, ∂_{0})^{−1}(κ_{1}(t_{0}), κ_{0}(t_{0})), there is a sequence p̃_{n} having two distinct limits p and p′, which is a contradiction.

Proposition 12. Let A be a locally path connected topological space, and (C, α) ∈ S(A). Then the continuous function (∂_{1}, ∂_{0}) : C_{1} ⟶ C_{0} × C_{0} is a covering map [5]. This means that each p ∈ C_{0} × C_{0} has an open neighborhood U with p ∈ U ⊂ C_{0} × C_{0} for which (∂_{1}, ∂_{0})^{−1}(U) is a disjoint union of subspaces U_{i} ⊂ C_{1}, each of which is mapped homeomorphically onto U by (∂_{1}, ∂_{0}).

Proof. We will prove this proposition in three steps. In the 1st step, for each p ∈ C_{0} we will find an open neighborhood U_{p} of p in C_{0} and an open neighborhood W_{p} of s(p) in C_{1} for which W_{p} is mapped homeomorphically onto U_{p} × U_{p} by (∂_{1}, ∂_{0}). In the 2nd step, for each (p, p′) ∈ C_{0} × C_{0} and q ∈ (∂_{1}, ∂_{0})^{−1}(U_{p} × U_{p}) we will find an open neighborhood V_{q} of q in C_{1}, which is mapped homeomorphically onto U_{p} × U′_{p} by (∂_{1}, ∂_{0}). And in the 3rd step we will prove that for each (p, p′) ∈ C_{0} × C_{0} and each q, q′ ∈ (∂_{1}, ∂_{0})^{−1}(U_{p} × U′_{p}) either V_{q} = V_{q′} or V_{q} ∩ V_{q′} = ∅.

1st step. Since (∂_{1}, ∂_{0}) is a local homeomorphism and C_{0} is locally path connected as well as A, it is possible to choose, for each p ∈ C_{0}, an open neighborhood W_{p} of s(p) ∈ C_{1} which is mapped homeomorphically onto U_{p} × U_{p}, where U_{p} is an open connected neighborhood of p in C_{0}. Let us prove that s(p) ∈ W_{p} for any p′ ∈ U_{p}. Denote by h_{p} the inverse homeomorphism of (∂_{1}, ∂_{0})|_{W_{p}} : W_{p} ⟶ U_{p} × U_{p}. Consider a path ω in C_{0} from p to p′. Then we have two paths s ∘ ω and h_{p}(ω, ω) in C_{1} with the common origin. By Proposition 11 s ∘ ω = h_{p}(ω, ω). Therefore s(p′) = sω(1) = h_{p}(ω(1), ω(1)) ∈ W_{p}.

2nd step. Suppose q ∈ (∂_{1}, ∂_{0})^{−1}(p, p′). Consider the continuous map

χ_{q} : U_{p} × U_{p′} ⟶ C_{1}, χ_{q}(p̃, p′) = h_{p′}(p′, p′) ∘ q ∘ (p̃, p);

here as above we mean by ∘ the internal composition in the internal category C.

Denote χ_{q}(U_{p} × U_{p′}) by V_{q}. By Proposition 11 it is clear that V_{q} is an open neighborhood of q which is mapped homeomorphically onto U_{p} × U_{p′} by (∂_{1}, ∂_{0}).

3rd step. V_{q} and V_{q′} are connected open neighborhoods; so this step follows immediately from Proposition 11.

Corollary 13. Let A be a locally path connected topological space and (C, α) ∈ S(A). Then the continuous function (∂_{1}, ∂_{0}) : C_{1} ⟶ C_{0} × C_{0} is a fibration in the sense of Hurevich.

Proof. This follows from Proposition 12 and [5, Theorem 2.33].

Example 2. Consider the case with E as a topos of presheaves on a small category C. Denote this topos as Sets^{C}. Then for each n ≥ 0, \underbrace{Sets^{C} ×_{Sets} ⋯ ×_{Sets} Sets^{C}}_{n+1} = Sets^{A^{n+1}} [3], any face and degeneracy geomet-

ric morphism from Sets^{C^{n+1}} to Sets^{C^{m+1}} is induced by the corresponding internal face and degeneracy functors in the natural simplicial internal category over Sets^{C}, and any geometric morphism Sets^{C′} ⟶ Sets^{C′′} is a local homeomorphism if and only if it is induced by a discrete fibration C′ ⟶ C′′. Therefore the category S(Sets^{C}) can be represented as the category of pairs (P, ϕ), where P is an internal category in Cat and ϕ is an internal functor from P to the antidiscrete internal category ad(C) such that ϕ_{0} : P_{0} ⟶ ad(C)_{0} = C and ϕ_{1} : P_{1} ⟶ ad(C)_{1} = C × C are discrete fibrations. Let us denote this new representation of the category S(Sets^{C}), i.e., the representation by internal categories and functors in Sets^{C}, simply by S(C).

Internal categories in Cat are called double categories [1]. Any double category D can be represented as a structure with objects, vertical morphisms, horizontal morphisms, and double morphisms (or cells). The vertical morphisms form a category whose composition is denoted by ∗ and identities by id. The horizontal structure also forms a category with composition denoted by ∘ and identities by 1. In fact, the whole structure can be described by the pasting of double morphisms. A double morphism has a horizontal domain and codomain, and a vertical domain and codomain. It can be pictured as:

\begin{matrix}A & ⟶^{h′} & B \\ _{v′}↑ & & ↑^{v} \\ C & ⟶^{h} & D \end{matrix}.

Definition 14. For any small category C, let S′(C) be the category whose objects are quadruples (F, p, Z, g), where p : F → C is a discrete fibration, Z is a small category, and g : F → Z is a functor, such that g_{0} : F_{0} → Z_{0} is a bijection and g_{1} : F_{1} → Z_{1} assigns an isomorphism in Z to each α ∈ F_{1}.

A morphism between (F, p, Z, g) and (F′, p′, Z′, g′) is a pair of functors (β : F → F′; γ : Z → Z′) such that p′ ∘ β = p and the diagram

F ⟶^{β} F′
g↓  ↓^{g′}
Z ⟶^{γ} Z′

is commutative.

Proposition 15. Let C be a small category. Then there is an equivalence of the categories

S(C) ⇆^{Ψ_{C}}_{Φ_{C}} S′(C)

and this equivalence is natural in C.

Proof. First let us construct the functor Φ_{C}.

Suppose (P, ϕ) ∈ S(C). Represent the double category P as a diagram:

\begin{matrix}P′′_{0} & ⇇ & P′′_{1} \\ ⇊ & & ⇊ \\ P′_{0} & ⇇ & P′_{1} \end{matrix} ,

where P′′_{0} ⇉ P′_{0} is the category P_{0} and P′′_{1} ⇉ P′_{1} is the category P_{1}, P′_{0} is the set of objects in P, P′′_{0} the set of vertical morphisms, P′_{1} the set of horizontal morphisms, and P′′_{1} the set of double morphisms or cells. The functors

ϕ_{0} = (ϕ′_{0}, ϕ′′_{0}) : (P′′_{0} ⇇ P′_{0}) ⟶ (C_{1} ⇇ C_{0}),

ϕ_{1} = (ϕ′_{1}, ϕ′′_{1}) : (P′′_{1} ⇇ P′_{1}) ⟶ (C_{1} × C_{1} ⇇_{∂_{0}×∂_{0}}^{∂_{1}×∂_{1}} C_{0} × C_{0})

are discrete fibrations.

Construct the functor g = (g_{0}, g_{1}) : (P′′_{0} ⇇ P′_{0}) ⟶ (P′_{0} ⇇ P′_{1}) as follows: Let g_{0} : P′_{0} ⟶ P′_{0} be the identity map 1_{P′_{0}}. For a vertical morphism α : x ⟶ y of P let g_{1}(α) be a horizontal mor-

phism λ such that there is a double morphism

\begin{matrix}x & ⟶^{λ} & y \\ ^{id_{x}}↑ & & ↑ \\ x & ⟶^{1_{x}} & x \end{matrix} .

Since ϕ_{0} and ϕ_{1} are discrete fibrations, such λ exists and is unique. Let us prove that g = (g_{0}, g_{1}) is a functor.

For each u ∈ P′_{0} and x ⟶^{α} y ⟶^{β} z α, β ∈ P′′_{0}, consider the diagrams

\begin{matrix}u & ⟶^{1_{u}} & u \\ ^{id_{u}}↑ & ^{id_{u}}↑ \\ u & ⟶^{1_{u}} & u \end{matrix} , \begin{matrix}x & ⟶^{g(α)} & y & ⟶^{g(β)} & z \\ ^{id_{x}}↑ & ^{id_{y}} ↑ & ^{β}↑ \\ x & ⟶^{g(α)} & y & ⟶^{1_{y}} & y \\ ^{id_{x}}↑ & ^{α}↑ & ^{α}↑ \\ x & ⟶^{1_{x}} & x & ⟶^{1_{x}} & x \end{matrix} .

Since in these diagrams all small squares determine cells in P, we have g(id_{u}) = 1_{u} and g(β ∘ α) = g(β) ∘ g(α). So g is a functor.

Now let us prove that for each (α : x ⟶ y) ∈ P′′_{0} g(α) is invertible. Consider the diagrams

\begin{matrix}x & ⟶^{g(α)} & y & ⟶^{λ′} & x \\ ^{id_{x}}↑ & & ↑^{α} & ↑^{id_{x}} \\ x & ⟶^{1_{x}} & x & ⟶^{1_{x}} & x \end{matrix} , \begin{matrix}x & ⟶^{λ′} & y & ⟶^{g(α)} & x \\ ^{α}↑ & & ↑^{id_{x}} & ↑^{α} \\ x & ⟶^{1_{x}} & x & ⟶^{1_{x}} & x \end{matrix}

in which all squares represent cells in P. Since ϕ_{1} is a discrete fibration, λ′ is the inverse of g(α)

Determine Φ_{C}(P, ϕ) to be the quadruple ((P′′_{0} ⇉ P′_{0}), ϕ_{0}, (P′_{1} ⇉ P′_{0}), g). Now let construct a functor Ψ_{C} : S′(C) ⟶ S(C). Suppose (F, p, Z, g) ∈ S′(C). We can identify objects of F with objects

of Z via the bijection p_{0} : F_{0} ⟶ Z_{0}. Determine a double category P as follows:

Let the set of objects of P be F_{0} ≅^{p_{0}} Z_{0}, the set of vertical morphisms F_{1}, the set of horizontal morphisms Z_{1}, and let the set of cells be the set of squares

\begin{matrix}u & ⟶^{γ} & z \\ ↑^{ϵ} & & ↑^{β} \\ x & ⟶^{α} & y \end{matrix} ,

where ϵ, β ∈ F_{1}, α, γ ∈ Z_{1}, and g(β) ∘ α = γ ∘ g(ϵ).

There is a naturally determined double functor ϕ : P ⟶ ad(C) which assigns p_{0}(x) to each x ∈ F_{0} ≅ Z_{0}, p_{1}(β) to each (β : x ⟶ y) ∈ F_{1}, (p_{0}(x), p_{0}(y)) ∈ C_{0} × C_{0} to each (α : x ⟶ y) ∈ Z_{1}, and (p_{1}(ϵ), p_{1}(β)) ∈ C_{1} × C_{1} to each cell

\begin{matrix}u & ⟶^{γ} & z \\ ↑^{ϵ} & & ↑^{β} \\ x & ⟶^{α} & y \end{matrix} .

We determine Ψ_{C}(F, p, Z, g)) to be the pair (P, ϕ).

One can easily check that the functors Φ_{C} and Ψ_{C} are inverse to each other.

The equivalences Φ_{C}, Ψ_{C} are clearly natural in C.

Remark 16. It is not difficult to see that the full subcategory of S′(C), which under the equivalence (Φ_{C}, Ψ_{C}) corresponds to the full subcategory of internal groupoids in S(C), consists of quadruples (F, p, Z, g) such that Z is a groupoid.

Remark 17. Define the forgetful functor T′_{C} : S(C)′ ⟶ Sets^{C} by

(F, p, Z, g) ↦ (F, p).

Then the diagram

[figure: S(C); S′_{C}(C); Sets^{C}; Φ_{C}; Ψ_{C}; T′_{C}; T_{C}]

is commutative.

Corollary 18. The category Cat Δ(Sets) is equivalent to the category of small categories and the category Grp Δ(Sets) is equivalent to the category of small groupoids.

Proof. The corollary follows from Example 2 when A is a one-point space, and also from Proposition 15 when C is a category with one object and one morphism.

3. A Fundamental Group

Definition 19. Let E ∈ Top. We will say that E admits the notion of a discrete category if the forgetful functor T_{E} : S(E) ⟶ E has a left adjoint F_{E} : E ⟶ S(E) such that T_{E} ∘ F_{E} ≅ 1_{E}. We will call this left adjoint functor F_{E} the discrete category functor for E, and internal categories in its range discrete internal categories.

Proposition 20. Let a topos E admit the notion of a discrete category; then the discrete category functor F_{E} : E ⟶ S(E) sends each X ∈ E to an internal groupoid in Δ(E), i.e., any “discrete category” in CatΔ(E)≅ S(E) is an internal groupoid.

Proof. We will prove this proposition in two steps. In the first step we will construct for each X ∈ E an internal equivalence η_{X} from F_{E}(X) to F_{E}^{op}(X), which is the identity on objects, and in the second step we will prove that the internal functor η_{X} “sends each morphism to its inverse.”

1st step. Since F_{E} is a left adjoint of T_{E}, there exists a function ϕ which assigns, to each pair of objects X ∈ E and Λ ∈ S(E), the bijection

ϕ = ϕ_{x,Λ} : Mor_{S(E)}(F (X); Λ) ⟶^{≈} Mor_{E}(X; T (Λ))  (7)

(we write T_{E} and F_{E} without the subscript E), and which is natural in X and Λ. Denote by η_{X} the internal functor (ϕ_{X,F_{(}X)}^{op})^{−1}(1_{X}) : F (X) ⟶ F^{op} (note that T F (X) = X).

Consider an internal functor η_{X}^{op} : F (X)^{op} ⟶ F (X). Since η_{X} is the identity on objects, η^{op} will also be the identity on objects. So ϕ_{X,F (X)}(η_{X}^{op}∘ η_{X}) = 1_{X}, and by (7) we have η_{X}^{op} ∘ η_{X} = 1_{F_{E}(X)}. Similarly, we prove η_{X} ∘ η_{X}^{op} = 1_{F_{E}(X)^{op}}.

2nd step. Let X ∈ E. Suppose F_{E}(X) = (X_{0}, X_{1}, ∂_{1}, ∂_{0}, s, m), where F_{E}(X) is represented as an object of Cat Δ(E). Consider the subobject Y of X_{1} in E ×_{Set} E, which is the “subobject of those morphisms ξ : ∂_{1}(ξ) ⟶

∂_{0}(ξ) in X_{1} for which ξ^{op} is the left inverse of ξ.” Such a subobject will be the limit of the diagram

[figure: (1_{E}, 1_{E})^{∗}(X); \underleftarrow{lim} (\begin{matrix}X_{1} & X_{1}^{∗} \\ ↘^{∂_{0}} & ^{∂_{1}^{∗}}↙ \\ (1_{E}, 1_{E})^{∗}(X_{0}) \end{matrix}); X_{1}; (pr_{0}, pr_{0})^{∗}(X_{1}); (pr_{0},pr_{1},pr_{0})^{∗}(m); pr^{∗}_{0}(s); ∂_{1}; \binom{1 η}{∂_{0}}]

Here ∂_{i}^{∗} and X_{1}^{∗} are the same as (pr_{1}, pr_{0})^{∗}(∂_{i}) and (pr_{1}, pr_{0})^{∗}(X_{1}).

Now let us construct a subcategory λ = (X, Y, ¯∂_{1}, ¯∂_{0}, s̄, m̄) of the internal category F_{E}(X) = (X, X_{1}, ∂_{1}, ∂_{0}, s, m) with a natural embedding i = (1_{X}, i_{1}) : λ ↣ F_{E}(X). We must determine the morphisms ¯∂_{1}, ¯∂_{0}, s̄, m̄ in such a way that they satisfy conditions (1′)–(6′) for an internal category and the following four conditions for an internal functor:

[figure: Y; X_{1}; δ^{1}(X); i_{1}; ¯∂_{1}; ∂_{1} ,]  (8)

[figure: Y; X_{1}; δ^{0}(X); i_{1}; ¯∂_{0}; ∂_{0} ,]  (9)

[figure: X; σ(Y); σ(X_{1}); σ(i_{1}) ,; s̄; s]  (10)

\begin{matrix}\underleftarrow{lim}( \begin{matrix}ρ_{2}^{0,1}(Y) ρ_{2}^{1,2}(Y) \\ ↘ ↙ \\ ρ_{2}^{1}(X) \end{matrix} ) & ⟶^{m̄} & ρ_{2}^{0,2}(Y) \\ ↓ & & ↓^{ρ_{2}^{0,2}(i_{1})} \\ \underleftarrow{lim}( \begin{matrix}ρ_{2}^{0,1}(X_{1}) ρ_{2}^{1,2}(X_{1}) \\ ↘ ↙ \\ ρ_{2}^{1}(X) \end{matrix} ) & ⟶^{m} & ρ_{2}^{0,2}(X_{1}). \end{matrix}  (11)

The commutativity of diagrams (8), (9) determines ¯∂_{1} and ¯∂_{0}, respectively. By the definition of Y it is clear that the existence of a morphism s̄ (resp., m̄) such that diagram (10) (resp., (11)) is commutative is equivalent to the commutativity of diagram (12) (resp., (13)):

[figure: X_{1}; σ(X_{1}); σ(X_{1}); σδ^{1}σ(X_{1}) ,; s; s; σ(α); σ(β)]  (12)

\underleftarrow{lim}( \begin{matrix}ρ_{2}^{0,1}(X_{1}) ρ_{2}^{1,2}(X_{1}) \\ ↘ ↙ \\ ρ_{2}^{1}(X) \end{matrix} )⟶^{m} ρ_{2}^{0,2}(X_{1})
↗  ↘ ^{ρ{0,2 \atop 2}(α)}
\underleftarrow{lim}( \begin{matrix}ρ_{2}^{0,1}(Y) ρ_{2}^{1,2}(Y) \\ ↘ ↙ \\ ρ_{2}^{1}(X) \end{matrix} )  ρ_{2}^{0,2}δ^{1}σ(X_{1})
↘  ↗ _{ρ{0,2 \atop 2}(β)}
\underleftarrow{lim}( \begin{matrix}ρ_{2}^{0,1}(X_{1}) ρ_{2}^{1,2}(X_{1}) \\ ↘ ↙ \\ ρ_{2}^{1}(X) \end{matrix} )⟶^{m} ρ_{2}^{0,2}(X_{1}) .

(13)

But the commutativity of diagrams (12), (13) as well as that of diagrams (1′)–(6′) for ¯∂_{1}, ¯∂_{0}, s̄, m̄ easily follow from conditions (1′)–(6′) for an internal category F (X) = (X, X_{1}, ∂_{1}, ∂_{0}, s, m).

So we have constructed an internal category λ and an internal inclusion i : λ ⟶ F_{E} such that i_{0} : λ_{0} = X ⟶ X = F_{E}(X)_{0} is the identity. But by the bijection ϕ_{X,λ} from (7) there exists an internal retraction j = ϕ_{X,λ} : F (X) ⟶ λ which is also the identity on objects. Consider the composition i∘j : F (X) ⟶ F (X) (note that i_{0} ∘j_{0} = 1_{X}). Then by (7) we have i∘j = 1. Therefore i_{1} ∘j_{1} = 1_{X_{1}}. So j_{1} is an epimorphism, but it is a monomorphism too by definition. As is wellknown [3], in any topos a morphism which is both mono- and epi- is an isomorphism. Therefore i_{1} is an isomorphism and hence λ is isomorphic to F (X) via i. So “every morphism in F_{E}(X) has a left inverse.” Similarly we prove that “every morphism in F_{E}(X) has the right inverse.” Hence F_{E}(X) is an internal groupoid.

Now we will define the fundamental group of those toposes which admit the notion of a discrete category. Suppose E is such a topos and θ is a point in it, i.e., θ is a geometric morphism θ : Sets ⟶ E. Consider the internal groupoid F_{E}(1) in Δ(E), where 1 is the terminal object in E. Then F_{E}(1)_{0} = 1 and therefore θ^{∗}(F_{E}(1)) is an internal groupoid in Sets (i.e., a small groupoid) which has only one object. We can consider the small groupoid θ^{∗}(F_{E}(1)) as a group. Elements of this group are morphisms in θ(F_{E}(1)) and the product is composition. Denote this group by π(E, θ). We will call the group π(E, θ) the fundamental group of the topos E at the point θ.

Let us prove that construction of the fundamental group from a topos which admit it is functorial. Suppose E_{1} and E_{2} are toposes which admit the notion of a discrete category, θ_{1} and θ_{2} are the points in them, respectively, and f is a geometric morphism f : E_{1} ⟶ E_{2} such that f ∘ θ_{1} = θ_{2}.

Consider two internal groupoids F_{E_{1}}(1_{E_{1}}) and f^{∗}(F_{E_{2}}(1_{E_{2}})) in Δ(E_{1}). Both have the terminal object in E_{1} as the object of objects. Therefore by bijection (7) there exists a unique internal functor f̃ from F_{E_{1}}(1_{E_{1}}) to f^{∗}(F_{E_{2}}(1_{E_{2}})). Consider the functor

θ_{1}^{∗}(f̃) : θ_{1}^{∗}(F_{E_{1}}(1_{E_{1}})) ⟶ θ_{1}^{∗}f^{∗}(F_{bfE_{2}}(1_{E_{2}})) = θ_{2}^{∗}(F_{E_{2}}(1_{E_{2}})).

This functor determines the group homomorphism

π(E_{1}, θ_{1}) ⟶ π(E_{2}, θ_{2})

which we denote by π(f).

One can easily check that π(1_{E}) = 1_{π(E)}, and if the geometric morphisms f and g are composable, then π(f ∘ g) = π(f) ∘ π(g). So fundamental groups of toposes with points determine a functor from the category of pointed toposes admitting the notion of a discrete category to the category of groups.

Now we will consider two examples which continue the examples from Section 2.

Example 1. E is a topos of sheaves on a topological space.

Proposition 21. Let A be a locally compact, locally path connected, and locally simply connected topological space. Then the topos of sheaves over A admits the notion of a discrete category.

Proof. We will prove this proposition in two steps. In the first step we will construct a functor F_{A} : shv(A) ⟶ S(A) such that T_{A} ∘ F_{A} = 1_{shv(A)} and in the second step we will prove that F_{A} is left adjoint to T_{A}.

1st step. Suppose a local homeomorphism p : E ⟶ A is any object of shv(A). We must assign to (p : E ⟶ A) ∈ shv(A) a pair (P^{A}, ϕ^{A}), where P^{A} is a continuous category and ϕ^{A} : P^{A} ⟶ ad(A) is a continuous functor such that ϕ_{1}^{A} : P_{1}^{A} ⟶ A × A and ϕ_{0}^{A} : P_{0}^{A} ⟶ A are local homeomorphisms.

Let P^{A} without a topology (or with a discrete topology as a small category) be the fundamental groupoid π(E) of the space E as in [5]. Any object of π(E) is a point of E, and an arrow x ⟶ x′ of π(E) is a homotopy class of paths from x to x′. (Such a path f is a continuous function I ⟶ E with f(0) = x, f(1) = x′, while two paths f, g with the same end points x and x′ are homotopic, when there is a continuous function F : I × I ⟶ E with F (t, 0) = f(t), F (t, 1) = g(t) and F (0, s) = x, F (1, s) = x′ for all s and t in I.) The composite of paths g : x′ ⟶ x′′ and f : x ⟶ x′ is the path h which is “f followed by g.” Composition applies also to homotopy classes and makes π(E) a category and a groupoid. (The inverse of any path is the same path traced in the opposite direction.)

Now determine topologies on P_{1}^{A} and P_{0}^{A}. Such a construction of the topological fundamental groupoid is considered in [6]. Since P_{0}^{A} as a set is the same as E, let a topology in P_{0}^{A} be the same as in E. (i.e., P_{0}^{A} = E as a topological space).

For any open subsets U and V in E, and any path ω in E also with ω(0) ∈ U, ω(1) ∈ V, consider the following subset of P_{1}^{A}:

< U, ω, V >= {ω_{2} ∗ ω ∗ ω_{1}|ω_{1} is a path in U and ω_{2} is a path in V }

(here by < κ > is denoted the homotopy class of a path κ and ∗ denotes the composition in the groupoid P_{A}).

The subsets < U, ω, V > determine an open base in P_{1}^{A}. Using this open base we will generate a topology in P_{1}^{A}. Equipped with these topologies, P^{A} becomes a continuous groupoid. (It is easy to check that the maps

∂_{1}, ∂_{0} : P_{1}^{A} ⟶ P_{0}^{A} = E, s : E ⟶ P_{1}^{A}, and m : \underleftarrow{lim}(P_{1}^{A} ⟶^{∂_{0}} E ⟵^{∂_{1}} O_{1}^{A}) ⟶

P_{1}^{A} are continuous.) There is a naturally determined continuous functor: ϕ^{A} : P^{A} ⟶ ad(a) ϕ_{1}^{A} = (p × p) ∘ (∂_{1}, ∂_{0}) ϕ_{0}^{A} = p. It remains to prove that ϕ_{1}^{A} is a local homeomorphism (ϕ_{0}^{A} = p is a local homeomorphism because (p : E ⟶ A) ∈ shv(A)). For this it is sufficient to prove that (∂_{1}, ∂_{0}) is a local homeomorphism because p × p is. First we prove that (∂_{1}, ∂_{0}) is an open map and then prove that (∂_{1}, ∂_{0}) is a local homeomorphism.

Any open W ⊂ P_{1}^{A} can be represented as a union W = ∪_{i}< U_{i}, ω_{i}, V_{i} >,

where U_{i} and V_{i} are linear connected open subsets. (Such a representation is possible because the space E is locally path connected as well as A.) Then (∂_{1}, ∂_{0})(W) = ∪_{i}U_{i} × V_{i} and therefore is open, i.e., (∂_{1}, ∂_{0}) is an open map.

For any < ω >∈ P_{1}^{A}, choose an open neighborhood < U, ω, V > of < ω >, where U and V are path connected and simply connected open sets. (This choice is also possible because E is a locally simply connected space as well as A.) It is easy to check that (∂_{1}, ∂_{0})|_{<U,ω,V >} :< U, ω, V >⟶ U × V is a bijection and also a continuous open map. Therefore (∂_{1}, ∂_{0}) is a local homeomorphism.

So we have constructed the object (P^{A}, ϕ^{A}) ∈ S(A). Let F_{A}(p : E ⟶ A) be (P^{A}, ϕ^{A}). This construction clearly implies that T_{A} ∘ F_{A} = 1_{shv(A)}.

2nd step. Suppose (p : E ⟶ A) ∈ shv(A) and (Λ, ψ) ∈ S(A). Consider the function

ϕ : Mor_{S(A)}(F_{A}(p : E ⟶ A); (Λ, ψ)) ⟶ Mor_{shv(A)}((p : E ⟶ A); Λ_{0}), (f : F_{A}(p : E ⟶ A) ⟶ Λ) ↦ (f_{0} : E ⟶ Λ_{0}).

We must prove that ϕ is a bijection. For this let us construct its inverse ϕ^{−1}. Suppose we are given f_{0} : E ⟶ Λ_{0} with ψ_{0} ∘ f_{0} = p and an element < ω > in F_{A}(p : E ⟶ A)_{1} (here ω is a path in E). Consider the following path κ in F_{A}(p : E ⟶ A)_{1}:

κ : I ⟶ F_{A}(p : E ⟶ A)_{1}, t ↦ κ(t) =< I ⟶ E_{s↦ω(st)} > .

The source of this path is the homotopy class s(ω(0)) ∈ F_{A}(p : E ⟶ A)_{1}, and the target is the homotopy class of ω. The image of the path κ under the map (∂_{1}, ∂_{0}) : F_{A}(p : E ⟶ A)_{1} ⟶ E × E is the pair of paths (ω(0), ω). The path κ is the lifting of the path (ω(0), ω) with the source s(< ω(0) >) ∈ F_{A}(p : E ⟶ A)_{1}. Therefore consider the path (f_{0} ∘ ω(0), f_{0} ∘ ω) in Λ_{0} × Λ_{0} and the lifting of this path to Λ_{1} via (∂_{1}, ∂_{0}) with a source s(f_{0} ∘ω(0)) ∈ Λ_{1}. (Such a lifting is possible and is unique by Proposition 13.) We determine ϕ^{−1}(f_{0})(< ω >) as the target of the lifted path. It is easy to check that ϕ^{−1}(f_{0}) so defined is a morphism in S(A) and ϕ^{−1} is the inverse of ϕ.

Let us determine what π(shv(A), θ) will be for a given point θ in shv(A) (A is a topological space with properties from Proposition 21). As is wellknown (see [4]), if A is sufficiently separated (for instance, if A is Hausdorff), then θ induced by a continuous map θ̃ : {∗} ⟶ A ({∗} is a one point space with the unique point ∗. In this case π(shv(A), θ) is a group of automorphisms of the object θ̃(∗) ∈ A in the fundamental groupoid π(A), i.e., π(shv(A), θ) = π(A, θ̃(∗)) (here π(A, θ̃(∗)) is the ordinary fundamental group of the space A at the point θ̃(∗)).

Example 2. E is a topos of presheaves.

Proposition 22. Let C be a small category; then the topos Sets^{C} admits the notion of a discrete category.

Proof. It is easily seen from [2] that the left adjoint functor of the forgetful functor T_{C} : S_{1}(C) ⟶ Sets^{C} exists and has the shape

F_{C} : Sets^{C} ⟶ S_{1}(C),
F_{C} : (p : Q ⟶ C) ↦ (Q, p, Z, g),

where Z is the category of fractions or the universal groupoid of Q and g is the natural functor from Q to Z. (Note that Q and Z have the same objects and g is identical on objects.)

The condition T_{C} ∘ F_{C} = 1_{Sets}^{C} is satisfied trivially.

Now let us determine what π(Sets^{C}, θ) will be for a given point θ in Sets^{C}. Suppose G is the category of fractions of C and i : C ⟶ G is a natural embedding. From the construction of the functor F_{C} it follows that i^{∗}(F_{G}(1) = i^{∗}(G, 1_{G}, G, 1_{G}) = (C, 1_{C}, G, i) = F_{C}(1). Therefore π(Sets^{C}, θ) = π(Sets^{G}, ĩ∘ θ); here by ĩ is denoted the geometric morphism from Sets^{C} to Sets^{G} induced by i.

By Diaconescu’s theorem [3] the point θ : Sets ⟶ Sets^{C} determines the flat presheaf θ̄ : C^{op} ⟶ Sets. From the definition of flat presheaves it follows that the value of the functor θ̄ is a nonempty set only on the objects of exactly one connected component of C^{op}. Denote this connected component by C_{0}^{op}. Then the group π(Sets^{C}, θ) = π(Sets^{G}, ĩ ∘ θ) will be isomorphic to the group of automorphisms of some object of C_{0} in the groupoid G.

Acknowledgement

The study is supported by ISF Grant MXH200 and INTAS Grant # 93-2618. I wish to thank Dr. M. Jibladze for valuable comments, in particular, when defining the category S(E).

References

1. R. Dawson and R. Pare, General associativity and composition for double categories. Cah. Topol. & Géom. Différ. Catégor. XXXIV(1993), No. 1, 57–79.

2. P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory. Springer-Verlag, Berlin etc., 1967.

3. P. T. Johnstone, Topos theory. Academic Press, London etc., 1977.

4. S. Mac Lane and I. Moerdijk, Sheaves in geometry and logic – A first introduction to topos theory. Springer-Verlag, Berlin etc., 1992.

5. E. H. Spanier, Algebraic topology. McGraw-Hill, New York, 1966.

6. R. Brown, Topology: a geometrical account of general topology, homology types and the fundamental groupoids. Ellis Horwood, New York etc., 1988.

(Received 28.11.1995)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia