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Some unsolved problems about the classifying topos for Boolean algebras, as well as about the axiomatic arithmetic of finite combinatorial toposes, are closely connected with some simple distinctions between finite automata.
@article{TAC_2004_13_a9, author = {F. William Lawvere}, title = {Functorial concepts of complexity for finite automata}, journal = {Theory and applications of categories}, pages = {164--168}, publisher = {mathdoc}, volume = {13}, year = {2004}, language = {en}, url = {http://geodesic.mathdoc.fr/item/TAC_2004_13_a9/} }
F. William Lawvere. Functorial concepts of complexity for finite automata. Theory and applications of categories, The Carboni Festschrift, Tome 13 (2004), pp. 164-168. http://geodesic.mathdoc.fr/item/TAC_2004_13_a9/