An Elementary Illustrated Introduction to Simplicial Sets
by Greg Friedman · arXiv / Rocky Mountain Journal of Mathematics
Expository paper explaining how simplicial sets generalise simplicial complexes, covering face and degeneracy maps, geometric realisation, and the simplicial route into homotopy theory. Illustrated throughout, written for readers who already know basic algebraic topology.
More resources on Simplicial Complexes
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
nLab: Simplicial Complex
Community-maintained nLab reference entry defining simplicial complexes combinatorially, with examples from graphs, nerves, posets and buildings. It contrasts them with simplicial sets, treats them as presheaves on finite sets, and covers geometric realisation and triangulable spaces.
Algebraic Topology
Learn simplicial complexes with N J Wildberger's Algebraic Topology course! Explore this fascinating area of math.
Hatcher's Algebraic Topology (free PDF)
Allen Hatcher's standard graduate textbook, freely downloadable from Cornell, covering fundamental groups, homology, cohomology and homotopy theory, with CW complexes introduced in Chapter 0 and used throughout. Readers learn to build spaces from cells and compute their algebraic invariants.
Computational Topology for Data Analysis
Cambridge University Press text whose free author-hosted pre-publication PDF treats simplicial complexes as computable data structures: Rips, Čech and Delaunay constructions, boundary matrices, matrix-reduction homology algorithms, then persistence, Reeb graphs and discrete Morse theory.
C3.9 Computational Algebraic Topology (Oxford course notes and videos)
Oxford graduate course with eight weeks of lecture notes and recorded videos, starting from simplicial complexes, geometric realisation and simplicial maps, then nerves, carriers, homology computation, cohomology, persistent homology, cellular sheaves and discrete Morse theory.