F. William Lawvere - Are homotopy types the same as infinitesimal skeleta?

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F. William Lawvere - Are homotopy types the same as infinitesimal skeleta?

	F. WILLIAM LAWVERE, Department of Mathmatics, SUNY at Buffalo, Buffalo, New York 14214, USA
	Are homotopy types the same as infinitesimal skeleta?

Among the functors ${\cal S} \to {\cal X}$ between cartesian-closed categories which have both left and right adjoints, there are those special ones in which those two adjoints are isomorphic, i.e. in which so to speak every component of an object X contains a unique point of X. For example, $\cal S$ could be the classifying topos for algebraic field extensions of a given field while ${\cal X}$ is the classifying topos for strongly local algebras over the same base; or ${\cal X}$ could be the category of presheaves for a category in $\cal S$ which has a terminal object which is also initial. Such could obviously arise by taking a suitable part ${\cal X} =\textrm{Infl}({\cal Z}) \to {\cal Z}$ of a large ${\cal Z}$ on which the two adjoints do not necessarily agree. But an opposite way to get from ${\cal Z}$ to such a special ${\cal X}$ arises by the Hurewicz construction for the case where the left adjoint from ${\cal Z}$ to $\cal S$ preserves finite products: the resulting re-enrichment ${\cal Z} \to {\cal H}_o({\cal Z}) ={\cal X}$ will satisfy our special equation if the left adjoint actually preserves infinite ( $\cal S$ -indexed) products, which is to be expected if components can be detected using continuous intervals, but not if only combinatorial intervals are available in ${\cal Z}$ .Thus, although considerable fine-tuning of the setting (over fixed $\cal S$ ) is still needed, it appears that the answer to the question in the title is: No, they are identical opposites.

Next: Michael Makkai - Weak Up: Category Theory / Théorie Previous: André Joyal - Title

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