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There are numerous expositions of simplicial homology in the literature.

Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes.

Hatcher in “Algebraic Topology” develops the homology theory of Δ-complexes, or, equivalently, semisimplicial sets.

Neither formalism is as efficient as simplicial sets:

• one needs 14 triangles to encode the 2-torus as a simplicial complex, whereas 2 triangles are sufficient for Δ-complexes or simplicial sets;

• homology of simplicial complexes requires one to introduce orientations on simplices, which is not necessary for simplicial sets;

• products of Δ-complexes are cumbersome to develop and work with, whereas products of simplicial sets are easy to construct as categorical products, whose presence further simplifies the resulting theory compared to Δ-complexes (e.g., the universal property of products, etc.).

Goerss and Jardine in “Simplicial Homotopy Theory” only spend a few lines defining the non-normalized Moore complex before Example I.1.4. They also explicitly assume knowledge of singular homology in that paragraph and do not compute any examples.

Gelfand and Manin in §I.4 of “Methods of Homological Algebra” give some basic definitions, but they use non-normalized chains and do not compute any examples. Greg Friedman's article “An elementary illustrated introduction to simplicial sets” has the same issue.

Finally, there are numerous expositions of singular homology of topological spaces, but these expositions typically have a different focus, and are not perfectly suitable for learning about homology of simplicial sets, since normalization of simplicial chains is not mentioned, but is crucial for computing the homology of simple examples with simplicial sets, e.g., surfaces, spheres, real projective spaces, etc.

Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology? To clarify, in this expository account normalized chains should be used to compute at least one simple example of homology, e.g., a 2-torus, real projective plane, etc.

There are numerous expositions of simplicial homology in the literature.

Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes.

Hatcher in “Algebraic Topology” develops the homology theory of Δ-complexes, or, equivalently, semisimplicial sets.

Neither formalism is as efficient as simplicial sets:

• one needs 14 triangles to encode the 2-torus as a simplicial complex, whereas 2 triangles are sufficient for Δ-complexes or simplicial sets;

• homology of simplicial complexes requires one to introduce orientations on simplices, which is not necessary for simplicial sets;

• products of Δ-complexes are cumbersome to develop and work with, whereas products of simplicial sets are easy to construct as categorical products, whose presence further simplifies the resulting theory compared to Δ-complexes (e.g., the universal property of products, etc.).

Goerss and Jardine in “Simplicial Homotopy Theory” only spend a few lines defining the non-normalized Moore complex before Example I.1.4. They also explicitly assume knowledge of singular homology in that paragraph and do not compute any examples.

Gelfand and Manin in §I.4 of “Methods of Homological Algebra” give some basic definitions, but they use non-normalized chains and do not compute any examples. Greg Friedman's article “An elementary illustrated introduction to simplicial sets” has the same issue.

Finally, there are numerous expositions of singular homology of topological spaces, but these expositions typically have a different focus, and are not perfectly suitable for learning about homology of simplicial sets, since normalization of simplicial chains is not mentioned, but is crucial for computing the homology of simple examples with simplicial sets, e.g., surfaces, spheres, real projective spaces, etc.

Is there an expository account of homology of simplicial sets that does not assume prior familiarity (as a formal dependence or as a source of intuition) with any variant of homology? To clarify, in this expository account normalized chains should be used to compute at least one simple example of homology, e.g., a 2-torus, real projective plane, etc.

There are numerous expositions of simplicial homology in the literature.

Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes.

Hatcher in “Algebraic Topology” develops the homology theory of Δ-complexes, or, equivalently, semisimplicial sets.

Neither formalism is as efficient as simplicial sets:

• one needs 14 triangles to encode the 2-torus as a simplicial complex, whereas 2 triangles are sufficient for Δ-complexes or simplicial sets;

• homology of simplicial complexes requires one to introduce orientations on simplices, which is not necessary for simplicial sets;

• products of Δ-complexes are cumbersome to develop and work with, whereas products of simplicial sets are easy to construct as categorical products, whose presence further simplifies the resulting theory compared to Δ-complexes (e.g., the universal property of products, etc.).

Goerss and Jardine in “Simplicial Homotopy Theory” only spend a few lines defining the non-normalized Moore complex before Example I.1.4. They also explicitly assume knowledge of singular homology in that paragraph and do not compute any examples.

Gelfand and Manin in §I.4 of “Methods of Homological Algebra” give some basic definitions, but they use non-normalized chains and do not compute any examples.

Finally, there are numerous expositions of singular homology of topological spaces, but these expositions typically have a different focus, and are not perfectly suitable for learning about homology of simplicial sets, since normalization of simplicial chains is not mentioned, but is crucial for computing the homology of simple examples with simplicial sets, e.g., surfaces, spheres, real projective spaces, etc.

Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology? To clarify, in this expository account normalized chains should be used to compute at least one simple example of homology, e.g., a 2-torus, real projective plane, etc.

There are numerous expositions of simplicial homology in the literature.

Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes.

Hatcher in “Algebraic Topology” develops the homology theory of Δ-complexes, or, equivalently, semisimplicial sets.

Neither formalism is as efficient as simplicial sets:

• one needs 14 triangles to encode the 2-torus as a simplicial complex, whereas 2 triangles are sufficient for Δ-complexes or simplicial sets;

• homology of simplicial complexes requires one to introduce orientations on simplices, which is not necessary for simplicial sets;

• products of Δ-complexes are cumbersome to develop and work with, whereas products of simplicial sets are easy to construct as categorical products, whose presence further simplifies the resulting theory compared to Δ-complexes (e.g., the universal property of products, etc.).

Goerss and Jardine in “Simplicial Homotopy Theory” only spend a few lines defining the non-normalized Moore complex before Example I.1.4. They also explicitly assume knowledge of singular homology in that paragraph and do not compute any examples.

Gelfand and Manin in §I.4 of “Methods of Homological Algebra” give some basic definitions, but they use non-normalized chains and do not compute any examples. Greg Friedman's article “An elementary illustrated introduction to simplicial sets” has the same issue.

Finally, there are numerous expositions of singular homology of topological spaces, but these expositions typically have a different focus, and are not perfectly suitable for learning about homology of simplicial sets, since normalization of simplicial chains is not mentioned, but is crucial for computing the homology of simple examples with simplicial sets, e.g., surfaces, spheres, real projective spaces, etc.

Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology? To clarify, in this expository account normalized chains should be used to compute at least one simple example of homology, e.g., a 2-torus, real projective plane, etc.

There are numerous expositions of simplicial homology in the literature.

Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes.

Hatcher in “Algebraic Topology” develops the homology theory of Δ-complexes, or, equivalently, semisimplicial sets.

Neither formalism is as efficient as simplicial sets:

• one needs 14 triangles to encode the 2-torus as a simplicial complex, whereas 2 triangles are sufficient for Δ-complexes or simplicial sets;

• homology of simplicial complexes requires one to introduce orientations on simplices, which is not necessary for simplicial sets;

• products of Δ-complexes are cumbersome to develop and work with, whereas products of simplicial sets are easy to construct as categorical products, whose presence further simplifies the resulting theory compared to Δ-complexes (e.g., the universal property of products, etc.).

Goerss and Jardine in “Simplicial Homotopy Theory” only spend a few lines defining the non-normalized Moore complex before Example I.1.4. They also explicitly assume knowledge of singular homology in that paragraph and do not compute any examples.

Finally, there are numerous expositions of singular homology of topological spaces, but these expositions typically have a different focus, and are not perfectly suitable for learning about homology of simplicial sets, since normalization of simplicial chains is not mentioned, but is crucial for computing the homology of simple examples with simplicial sets, e.g., surfaces, spheres, real projective spaces, etc.

Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology? To clarify, in this expository account normalized chains should be used to compute at least one simple example of homology, e.g., a 2-torus, real projective plane, etc.

There are numerous expositions of simplicial homology in the literature.

Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes.

Hatcher in “Algebraic Topology” develops the homology theory of Δ-complexes, or, equivalently, semisimplicial sets.

Neither formalism is as efficient as simplicial sets:

• one needs 14 triangles to encode the 2-torus as a simplicial complex, whereas 2 triangles are sufficient for Δ-complexes or simplicial sets;

• homology of simplicial complexes requires one to introduce orientations on simplices, which is not necessary for simplicial sets;

• products of Δ-complexes are cumbersome to develop and work with, whereas products of simplicial sets are easy to construct as categorical products, whose presence further simplifies the resulting theory compared to Δ-complexes (e.g., the universal property of products, etc.).

Goerss and Jardine in “Simplicial Homotopy Theory” only spend a few lines defining the non-normalized Moore complex before Example I.1.4. They also explicitly assume knowledge of singular homology in that paragraph and do not compute any examples.

Gelfand and Manin in §I.4 of “Methods of Homological Algebra” give some basic definitions, but they use non-normalized chains and do not compute any examples.

Finally, there are numerous expositions of singular homology of topological spaces, but these expositions typically have a different focus, and are not perfectly suitable for learning about homology of simplicial sets, since normalization of simplicial chains is not mentioned, but is crucial for computing the homology of simple examples with simplicial sets, e.g., surfaces, spheres, real projective spaces, etc.

Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology? To clarify, in this expository account normalized chains should be used to compute at least one simple example of homology, e.g., a 2-torus, real projective plane, etc.