In category theory, a discipline within mathematics, the nerve N(C) of a small category C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space, called the classifying space of the category C. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory.
Motivation The nerve of a category is often used to construct topological versions of moduli spaces. If X is an object of C, its moduli space should somehow encode all objects isomorphic to X and keep track of the various isomorphisms between all of these objects in that category. This can become rather complicated, especially if the objects have many non-identity automorphisms. The nerve provides a combinatorial way of organizing this data. Since simplicial sets have a good homotopy theory, one can ask questions about the meaning of the various homotopy groups πn(N(C)). One hopes that the answers to such questions provide interesting information about the original category C, or about related categories. The notion of nerve is a direct generalization of the classical notion of classifying space of a discrete group; see below for details.
Construction Let C be a small category. There is a 0-simplex of N(C) for each object of C. There is a 1-simplex for each morphism f : x → y in C. Now suppose that f: x → y and g : y → z are morphisms in C. Then we also have their composition gf : x → z. The diagram suggests our course of action: add a 2-simplex for this commutative triangle. Every 2-simplex of N(C) comes from a pair of composable morphisms in this way. The addition of these 2-simplices does not erase or otherwise disregard morphisms obtained by composition, it merely remembers that this is how they arise. In general, N(C)k consists of the k-tuples of composable morphisms
A 0 → A 1 → A 2 → ⋯ → A k − 1 → A k {\displaystyle A_{0}\to A_{1}\to A_{2}\to \cdots \to A_{k-1}\to A_{k}}
of C. To complete the definition of N(C) as a simplicial set, we must also specify the face and degeneracy maps. These are also provided to us by the structure of C as a category. The face maps
d i : N ( C ) k → N ( C ) k − 1 {\displaystyle d_{i}\colon N(C)_{k}\to N(C)_{k-1}}
are given by composition of morphisms at the ith object (or removing the ith object from the sequence, when i is 0 or k). This means that di sends the k-tuple
A 0 → ⋯ → A i − 1 → A i → A i + 1 → ⋯ → A k {\displaystyle A_{0}\to \cdots \to A_{i-1}\to A_{i}\to A_{i+1}\to \cdots \to A_{k}}
to the (k − 1)-tuple
A 0 → ⋯ → A i − 1 → A i + 1 → ⋯ → A k . {\displaystyle A_{0}\to \cdots \to A_{i-1}\to A_{i+1}\to \cdots \to A_{k}.}
That is, the map di composes the morphisms Ai−1 → Ai and Ai → Ai+1 into the morphism Ai−1 → Ai+1, yielding a (k − 1)-tuple for every k-tuple. Similarly, the degeneracy maps
s i : N ( C ) k → N ( C ) k + 1 {\displaystyle s_{i}:N(C)_{k}\to N(C)_{k+1}}
are given by inserting an identity morphism at the object Ai. Simplicial sets may also be regarded as functors Δop → Set, where Δ is the category of totally ordered finite sets and order-preserving morphisms. Every partially ordered set P yields a (small) category i(P) with objects the elements of P and with a unique morphism from p to q whenever p ≤ q in P. We thus obtain a functor i from the category Δ to the category of small categories. We can now describe the nerve of the category C as the functor Δop → Set
N ( C ) ( _ ) = F u n ( i ( _ ) , C ) . {\displaystyle N(C)(\_)=\mathrm {Fun} (i(\_),C).\,}
This description of the nerve makes functoriality transparent; for example, a functor between small categories C and D induces a map of simplicial sets N(C) → N(D). Moreover, a natural transformation between two such functors induces a homotopy between the induced maps. This observation can be regarded as the beginning of one of the principles of higher category theory. It follows that adjoint functors induce homotopy equivalences. In particular, if C has an initial or final object, its nerve is contractible.
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