Simplicial set
A simplicial set is a combinatorial structure in algebraic topology defined as a contravariant functor from the simplex category Δ—whose objects are finite ordered sets = {0, 1, ..., n} and whose morphisms are non-decreasing functions—to the category of sets, or equivalently, a sequence of sets X_n (for n \geq 0) equipped with face maps d_i: X_n \to X_{n-1} and degeneracy maps s_i: X_n \to X_{n+1} (for appropriate indices) satisfying the simplicial identities, including relations like d_i d_j = d_{j-1} d_i for i < j and compatibility conditions between faces and degeneracies.[1][2] Elements of X_n are called n-simplices, which generalize the simplices of geometric simplicial complexes but allow for degeneracies—simplices arising from lower-dimensional ones via degeneracy maps—enabling more flexible modeling of topological attachments without requiring distinct vertices or injective face inclusions.[1] Introduced by Samuel Eilenberg and J. A. Zilber in 1950 as "complete semi-simplicial complexes" to study singular homology, simplicial sets extend earlier notions from combinatorial topology and provide a purely algebraic framework for capturing homotopy-invariant properties of spaces.[3] In algebraic topology, simplicial sets play a central role through the adjunction between the geometric realization functor |\cdot|: \mathbf{sSet} \to \mathbf{Top}, which constructs a CW-complex from a simplicial set by gluing standard simplices along faces, and the singular complex functor \operatorname{Sing}: \mathbf{Top} \to \mathbf{sSet}, which maps a space to the simplicial set of its continuous simplices; this adjunction induces weak homotopy equivalences, allowing simplicial sets to model spaces up to homotopy.[1][2] Key subclasses include Kan complexes, which satisfy the horn-filling condition and model ∞-groupoids, and quasicategories (or ∞-categories), which further structure simplicial sets to encode higher-dimensional categorical compositions and homotopy coherences.[1] Simplicial sets facilitate computations in homotopy theory, such as simplicial homology and cohomology, and underpin model category structures on the category \mathbf{sSet}, enabling tools like fibrant replacements and derived functors without direct reference to underlying topological spaces.[2] Their influence extends to derived algebraic geometry and higher topos theory, where they serve as presentations for abstract homotopy types.[1]Introduction
Motivation
Simplicial sets emerged in the mid-20th century as a response to the need for a combinatorial framework in algebraic topology that could handle homotopy theory in a functorial manner, particularly during the 1940s and 1950s when researchers sought to unify and axiomatize homology and cohomology theories for abstract spaces and groups.[4] Pioneered by Samuel Eilenberg and J. A. Zilber in their 1950 work on semi-simplicial complexes, this approach was driven by developments in singular homology, where traditional simplicial complexes proved insufficient for capturing the full structure of topological spaces without rigid geometric constraints.[4] The motivation stemmed from the desire to model homotopy invariants algebraically, avoiding the complexities of continuous maps in singular chains while enabling precise computations of homology groups.[1] As a discrete, combinatorial alternative to singular chains or CW-complexes, simplicial sets provide a way to represent topological spaces through sets of simplices equipped with face and degeneracy maps, sidestepping embedding problems that arise when realizing abstract spaces geometrically.[5] Unlike CW-complexes, which rely on cell attachments that can be topologically irregular, simplicial sets emphasize purely algebraic data, making them ideal for functorial constructions and avoiding pathological issues like non-uniqueness in decompositions.[1] This combinatorial purity facilitates the study of homotopy without requiring explicit embeddings, as the structure encodes higher-dimensional relations directly through operators.[5] A key advantage over classical simplicial complexes lies in the inclusion of degenerate simplices, which allow the model to capture essential homotopy information that would otherwise be lost by restricting to non-degenerate elements alone.[5] In simplicial complexes, degeneracies are absent, limiting their ability to represent homotopy equivalences functorially; simplicial sets rectify this by incorporating degeneracy operators, enabling a faithful algebraic encoding of paths and higher homotopies.[1] This flexibility proved crucial for advancing homotopical algebra in the 1950s.[4] Simplicial sets also play a pivotal role in bridging category theory and topology through adjoint functors, such as the geometric realization functor that recovers topological spaces from simplicial data and the singular functor that assigns simplicial sets to spaces, forming a Quillen equivalence that preserves homotopical structure.[1] This functorial interplay, rooted in the category of simplicial sets as presheaves on the simplex category, underscores their utility in integrating categorical abstractions with topological insights.[1]Intuition
Simplicial sets can be intuitively understood as combinatorial structures that assemble abstract simplices into higher-dimensional objects, much like building polyhedra by gluing triangular faces along edges, but extended to allow for flexible deformations that capture homotopy-theoretic phenomena. At their core, these structures consist of collections of simplices organized by dimension, where lower-dimensional simplices serve as the boundaries or faces of higher ones, enabling a layered construction that mimics the topology of spaces without relying on explicit metric or embedding details.[1] Unlike traditional simplicial complexes, which enforce strict geometric constraints such as non-overlapping simplices and distinct vertices, simplicial sets incorporate additional "degenerate" simplices that arise from collapsing higher-dimensional elements onto lower ones, providing the flexibility needed to model retractions and paths in a homotopy-invariant way. These degenerate simplices allow the structure to represent not just rigid shapes but also the ways in which spaces can be continuously deformed, making simplicial sets a powerful tool for abstracting topological intuitions into purely algebraic terms. An n-simplex in this framework acts as an abstract "n-dimensional block," equipped with designated faces (lower-dimensional simplices) and vertices, where degeneracy operators permit dimensional reduction, such as turning an edge into a looped vertex to encode basepoints or fixed points.[1] This combinatorial approach draws a direct analogy to topological simplices: 0-simplices correspond to points or vertices, 1-simplices to directed edges connecting them, 2-simplices to filled triangles bounded by three edges, and higher n-simplices to their n-dimensional generalizations, all defined relationally through face inclusions rather than coordinates. Face and degeneracy maps serve as the precise gluing rules that enforce compatibility between these blocks, ensuring the overall structure behaves coherently across dimensions. For a concrete visualization, consider the simplicial circle, which models the circle S^1 as a single non-degenerate 1-simplex (the looping edge) glued to itself at endpoints via degenerate 0-simplices (repeated basepoints), effectively capturing the topology of a circle through this minimal combinatorial data.[1][5]Definition
Simplex Category
The simplex category \Delta has as objects the finite ordinals for $n \geq 0$, where $ = \{0 < 1 < \cdots < n\}$.[](https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/Goerss-Jardine2.pdf) The object ${{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}}$, consisting of a single element, serves as the terminal object in $\Delta$, as there is a unique morphism from any to {{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}}.[6] Morphisms in \Delta are the order-preserving maps between these ordinals, namely the non-decreasing functions f: \to such that i \leq j implies f(i) \leq f(j).[7] Among these morphisms, the injections are termed face maps, while the surjections are called degeneracy maps.[6] The category \Delta is skeletal, with its objects in canonical bijection with the natural numbers via the representatives $$, and it is generated by the face and degeneracy morphisms under composition, subject to the simplicial identities.[7] The isomorphisms in \Delta are precisely the identity maps, as any bijective order-preserving map on finite totally ordered sets must be the identity.[6] Simplicial sets are defined as contravariant functors from \Delta to the category of sets, or equivalently as presheaves on \Delta.[7]Simplicial Sets
A simplicial set is defined as a contravariant functor from the simplex category to the category of sets.[1] Specifically, given the simplex category \Delta, a simplicial set X is a functor X : \Delta^{\mathrm{op}} \to \mathrm{Set}, where \Delta^{\mathrm{op}} is the opposite category of \Delta.[8] For each object $$ in \Delta, which corresponds to the ordered set \{0 < 1 < \cdots < n\}, the set X(), denoted X_n, consists of the n-simplices of X.[1] The action of X on morphisms in \Delta^{\mathrm{op}} (equivalently, contravariant action on morphisms in \Delta) assigns to each order-preserving map \sigma : \to in \Delta a function X(\sigma) : X_n \to X_m.[9] Morphisms between simplicial sets are natural transformations between these functors.[1] A natural transformation \eta : X \to Y consists of a family of functions \eta_n : X_n \to Y_n for each n \geq 0, such that for every morphism \sigma : \to in \Delta, the diagram \begin{CD} X_n @>{\eta_n}>> Y_n \\ @V{X(\sigma)}VV @VV{Y(\sigma)}V \\ X_m @>>{\eta_m}> Y_m \end{CD} commutes.[8] The category of simplicial sets, denoted \mathrm{sSet} or \mathrm{Set}^{\Delta^{\mathrm{op}}}, has simplicial sets as objects and natural transformations as morphisms; it is the functor category from \Delta^{\mathrm{op}} to \mathrm{Set}.[9] Within a simplicial set X, an n-simplex in X_n is called degenerate if it lies in the image of one of the degeneracy maps s_i : X_{n-1} \to X_n for $0 \leq i \leq n-1, where these maps are induced by the degeneracy morphisms in \Delta.[1] Otherwise, it is non-degenerate. Every degenerate simplex arises uniquely from a lower-dimensional simplex via repeated applications of degeneracy maps, as guaranteed by the Eilenberg-Zilber lemma.[1] Non-degenerate simplices form a basis for the simplicial set in the sense that every simplex is a degeneracy of a unique non-degenerate one.[9] The initial object in \mathrm{sSet} is the empty simplicial set \emptyset, which assigns the empty set to every X_n and has no non-degenerate simplices.[8] The terminal object is the simplicial set \Delta^0 (or y(0), the representable functor at {{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}}), which has exactly one n-simplex for each n \geq 0, generated by degeneracies from the unique 0-simplex; it is the constant functor on the singleton set.[1]Face and Degeneracy Operators
In a simplicial set X, the face operators are maps d_i: X_n \to X_{n-1} for $0 \leq i \leq n, each induced by the corresponding coface morphism \delta^i: [n-1] \to in the simplex category \Delta, which is the order-preserving injection skipping the i-th position.[10][2] Similarly, the degeneracy operators are maps s_j: X_n \to X_{n+1} for $0 \leq j \leq n, induced by the codegeneracy morphism \sigma_j: [n+1] \to in \Delta, which is the order-preserving surjection that identifies the j-th and (j+1)-th positions by repeating the value at j.[10][2] These operators satisfy the simplicial identities, which ensure compatibility under composition:- d_i d_j = d_{j-1} d_i for i < j,
- s_i s_j = s_{j+1} s_i for i \leq j,
- d_i s_j = \mathrm{id} for i = j or i = j+1,
- d_i s_j = s_{j-1} d_i for i < j,
- d_i s_j = s_j d_{i-1} for i > j+1.