## Return to Answer

3 added 5 characters in body

If your homology theory is of the form $H_n(X) = H_n(S(X))$ where $S$ is some functor from your original category to non-negative chain complexes, then the Dold-Kan correspondence gives you a corresponding simplicial abelian group $\Gamma(S(X))$ and hence (by realization) a topological space in which you are "counting holes".

2 removed backticks on discovering they worsen display problem

If your homology theory is of the form $H_n(X) = H_n(S(X))$ H_n(S(X)) $where$S$is some functor from your original category to non-negative chain complexes, then the Dold-Kan correspondence gives you a corresponding simplicial abelian group$\Gamma(S(X))$\Gamma(S(X))$ and hence (by realization) a topological space in which you are "counting holes".

1

If your homology theory is of the form $H_n(X) = H_n(S(X))$ where $S$ is some functor from your original category to non-negative chain complexes, then the Dold-Kan correspondence gives you a corresponding simplicial abelian group $\Gamma(S(X))$ and hence (by realization) a topological space in which you are "counting holes".
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 10, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9076836705207825, "perplexity_flag": "middle"}