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expanded model structure on functors by adding a long list of properties
I have edited the Properties-section at model structure on functors to make more explicit that all of the statements about functorial dependen on domain and codomain hold also for the case of $\mathbf{S}$-enriched functors on $\mathbf{S}$-enriched categories.
(This was previously mentioned in between the lines, but not made sufficiently explicit.)
I have added pointer to
Robert Piacenza, Homotopy theory of diagrams and CW-complexes over a category, Can. J. Math. Vol 43 (4), 1991 (pdf)
also chapter VI of Peter May et al., Equivariant homotopy and cohomology theory, 1996 (pdf)
to the entries model structure on functors, Elmendorf’s theorem and Model categories of diagram spectra.
Why does HTT Proposition A.3.3.2 (existence of injective and projective model structures on enriched diagram categories) require $\mathbf{S}$ to be excellent? In particular, why does it require the “invertibility hypothesis”? The proof isn’t written out but said to be “identical to that of Proposition A.2.8.2”, the existence of injective and projective model structures on unenriched diagram categories; I glanced at the latter proof but wasn’t able to see where the invertibility hypothesis might be used.
Added Moser’s theorem about existence of both projective and injective model structures for all accessible model categories, which is much more general than Lurie’s (although it doesn’t imply that the injective model structure is cofibrantly generated, only that it is accessible).
moved this proposition (on relation to functor $\infty$-categories) out of the subsection “Properties – Functoriality in domain and codomain” (where it was misplaced, into “Properties – General”).
Also adjusted typesetting of arrows in the entry (replacing \leftarrow
by \longleftarrow
and \to
by \longrightarrow
in order for the adjunctions to display properly)
added (here) a brief paragraph stating the relation to $\infty$-functors (Prop. 4.2.4.4 in HTT).
$N \big( \mathbf{sFunc}(\mathbf{D}, \mathbf{C})^\circ \big) \;\; \underset{qCat}{\simeq} \;\; Func_\infty \Big( N(\mathbf{D}) ,\, N\big( \mathbf{C}^\circ \big) \Big) \,.$(We have corresponding and more discussion at (infinity,1)-category of (infinity,1)-functors, but it’s good to have a pointer here, too).
By the way: Is there a general naturality statement saying that this equivalence is suitably natural, e.g. compatible with precomposition operations $f^\ast$?
[edit: Have posted what looks like an answer to this question in another thread: here]
Re #10: There is a question about this on MathOverflow: https://mathoverflow.net/questions/440965/htt-remark-4-2-4-5
Thanks, I have added a footnote: there.
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