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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 8th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexRemoved a query box: How can $id^*$ be an identity functor when that is not contravariant? Maybe each $f^*$ is a covariant functor but the mapping $f \mapsto f^*$ is a contravariant functor? But then it\'s automatic that $id^* = id$ (and furthermore that $(f ; g)^* = f^* \circ g^*$). ---Toby [[David Roberts]]: whoops! I didn't pick that up. I think you are partly right: it should be some sort of contravariant assignment $f\mapsto f^*$, but maybe not functorial (since I believe that *category* should be replaced as I said below). The protoypical example, AFAICS, is assigning the category of locally homotopy trival fibrations over the given space. It is not spelled out in detail in the paper. <a href="https://ncatlab.org/nlab/revision/diff/fibration+theory/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/fibration+theory/6">v6</a>, <a href="https://ncatlab.org/nlab/show/fibration+theory">current</a>

   Removed a query box:

   How can $id^*$ be an identity functor when that is not contravariant? Maybe each $f^*$ is a covariant functor but the mapping $f \mapsto f^*$ is a contravariant functor? But then it's automatic that $id^* = id$ (and furthermore that $(f ; g)^* = f^* \circ g^*$). —Toby

   David Roberts: whoops! I didn’t pick that up. I think you are partly right: it should be some sort of contravariant assignment $f\mapsto f^*$, but maybe not functorial (since I believe that category should be replaced as I said below). The protoypical example, AFAICS, is assigning the category of locally homotopy trival fibrations over the given space. It is not spelled out in detail in the paper.

   diff, v6, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 8th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexRemoved a query box: What makes an open cover 'numerable'? ---Toby [[David Roberts]]: A cover is numerable if it admits a subordinate partition of unity. [[numerable open cover|Numerable open covers]] form a site. The axiom is there to link locally homotopically trivial fibrations and Dold fibrations (see theorem 2.3 in Wirth-Stasheff, due to Dold.) Also, the uniqueness should at least be demoted to unique-up-to-isomorphism. <a href="https://ncatlab.org/nlab/revision/diff/fibration+theory/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/fibration+theory/6">v6</a>, <a href="https://ncatlab.org/nlab/show/fibration+theory">current</a>

   Removed a query box:

   What makes an open cover ’numerable’? —Toby

   David Roberts: A cover is numerable if it admits a subordinate partition of unity. Numerable open covers form a site. The axiom is there to link locally homotopically trivial fibrations and Dold fibrations (see theorem 2.3 in Wirth-Stasheff, due to Dold.)

   Also, the uniqueness should at least be demoted to unique-up-to-isomorphism.

   diff, v6, current

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 8th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexRemoved a query box: [[David Roberts]]: The axioms are just copied from Wirth--Stasheff JHRS 1(1) 2006, p 273. They need to be clarified a little, as the notion of homotopy and homotopic are undefined. We could ask that $E(B)$ is a category of fibrant objects or a Quillen model category or $(\infty,1)$-category or a category with an interval objects or something. One could even ask for a subcategory of $Top$ which is closed under some conditions. In that instance, something needs to be said about the compatibility of homotopies etc with the functors $f^*$. _Toby_: I know that you\'re just copying things, so maybe you don\'t know the answers to my questions, but so far I don\'t even understand the parts that I should be able to understand! <a href="https://ncatlab.org/nlab/revision/diff/fibration+theory/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/fibration+theory/6">v6</a>, <a href="https://ncatlab.org/nlab/show/fibration+theory">current</a>

   Removed a query box:

   David Roberts: The axioms are just copied from Wirth–Stasheff JHRS 1(1) 2006, p 273. They need to be clarified a little, as the notion of homotopy and homotopic are undefined. We could ask that $E(B)$ is a category of fibrant objects or a Quillen model category or $(\infty,1)$-category or a category with an interval objects or something. One could even ask for a subcategory of $Top$ which is closed under some conditions. In that instance, something needs to be said about the compatibility of homotopies etc with the functors $f^*$.

   Toby: I know that you're just copying things, so maybe you don't know the answers to my questions, but so far I don't even understand the parts that I should be able to understand!

   diff, v6, current

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 8th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## References Original PhD thesis: * James Frederick Wirth. Fiber spaces and the higher homotopy cocycle relations. PhD thesis, University of Notre Dame, 1965. Modern treatment: * James Wirth, [[Jim Stasheff]], _Homotopy transition cocycles_ J. Homotopy Relat. Struct., 1(1):273–283, 2006. [EuDML](https://eudml.org/doc/128360), [arXiv](https://arxiv.org/abs/math/0609220). <a href="https://ncatlab.org/nlab/revision/diff/fibration+theory/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/fibration+theory/6">v6</a>, <a href="https://ncatlab.org/nlab/show/fibration+theory">current</a>

   Added:

   References

   Original PhD thesis:

   • James Frederick Wirth. Fiber spaces and the higher homotopy cocycle relations. PhD thesis, University of Notre Dame, 1965.

   Modern treatment:

   • James Wirth, Jim Stasheff, Homotopy transition cocycles J. Homotopy Relat. Struct., 1(1):273–283, 2006. EuDML, arXiv.

   diff, v6, current