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finally a stub for Segal condition. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).
added discussion to Segal condition of how it is not quite a sheaf condition on $\Delta$ but is related to a sheaf condition on some $\Delta_0$.
I have further expanded at Segal condition. In particular I wrote a section In terms of sheaf conditions (which maybe got a bit more detailed than is enjoyable, but anyway).
Let’s look at the upshot, I am wondering if this is supposed to be telling us something:
write
$\left\{1 \Leftarrow 0\right\} \; \stackrel{i}{\to} \; \Delta_0 \; \stackrel{j}{\to} \; \Delta$for the canonical inclusion functors, where $\Delta_0$ is the full subcategory of graphs on the $\Delta[n]$ regarded as graphs (so morphisms $\Delta[k_1] \to \Delta[k_2]$ in $\Delta_0$ send elementary edges to elementary edges: they just pick a $k_1$-long subequence in $k_2$. In particular they are all injections, both on vertices and on edges).
Then the statement is that
$\array{ Cat &\stackrel{N}{\hookrightarrow}& PSh(\Delta) \\ \downarrow^{\mathrlap{U}} && \downarrow^{\mathrlap{j^*}} \\ Graph \simeq Sh(\Delta_0) &\stackrel{i_*}{\hookrightarrow}& PSh(\Delta_0) }$is a pullback diagram (in the 1-category of categories).
But let’s think about this further: unless I am mixed up we have:
$i_*$ is the direct image of a geometric morphism;
$j^*$ is the direct image of a geometric morphism
all three toposes $Graph, PSh(\Delta_0), PSh(\Delta)$ are cohesive.
So this characterizes $Cat$ as a limit of a pullback diagram in the Image of $CohesiveTopos \hookrightarrow Topos \stackrel{F}{\to} Cat$. Of course this $F$ does not preserve pullbacks, and of course $Cat$ is not a topos. But it’s still curious that this way it is the pullback in $Cat$ of a diagram of cohesive toposes. (Unless I am making a mistake somewhere.) I am wondering if this is telling us something about how to speak about (pre-)category objects in cohesive homotopy type theory.
Added brief paragraphs
Somewhat telegraphic for the moment. Am running out of steam now.
We have no entry precategory.I understand this term as referring to a weighted (with the name of the arrow) directed graph where it is allowed that the same name is assigned to different arrows. Is this also intended in your naming?
edit: ok, precategory redirects to paracategory.
We have no entry precategory.I understand this term as referring to a weighted (with the name of the arrow)
Where Segal condition refers and points to “pre-categories” it points to pre-category object in an infinity-category (see there).
This refers to a simplicial object $X_\bullet$ in an $\infty$-category which does satisfy the Segal condition, but which does not necessarily satisfy the additional “completess” condition (saying that $Core(X_\bullet)$ is a homotopy-constant simplicial object or equivalently that $id \colon X_0 \to Equiv(X_1)$ is an equivalence).
Calling this a “pre-category object” is not standard, but it is what we ended up with after some discussion here. There is no really standard term for this at the moment. Elsewhere it is called just a “category object” and then with the other conidition imposed it would be a “complete Segal object”. But the idea here was that “complete Segal object” is an unfortunately undescriptive term for such an important concept.
But, yeah, eventually we might want to settle on something else. We had had plenty of discussion on this elsewhere, not entirely conclusive maybe.
No, not conclusive, although since as Stephan points out the word “precategory” is used for some other very different things, perhaps we should change it to something which is at least unambiguous.
The notation in Definition 2 at Segal condition seems to be not in line with that on simplex category - what in the first article is denoted by $\Delta[n]$ would have been denoted by $[n]$ in the latter article.
Stephan, I’m not sure of a good polite way to say this, but: that’s bound to happen with different people working on different articles. Does it bother you that there are different notations? The meaning seems clear enough in either case, and striving for absolute consistency of notation across the nLab seems close to impossible (and in many cases, including this one IMO, not worth worrying about).
I have added a further Remark on how the pullback diagram that characterizes $Cat$ as in Prop. 6 involves a bunch of adjoints
$\array{ Cat &\stackrel{\overset{\tau}{\leftarrow}}{\underoverset{N}{\bottom}{\hookrightarrow}}& PSh(\Delta) \simeq sSet \\ {}^{\mathllap{U}}\downarrow \vdash \uparrow^{\mathrlap{F}} && {}^{\mathllap{j^*}}\downarrow \vdash \uparrow^{\mathrlap{j_!}} \\ Graph \simeq Sh(\Delta_0) &\stackrel{\overset{i^*}{\leftarrow}}{\underoverset{i_*}{\bottom}{\hookrightarrow}}& PSh(\Delta_0) }$and how in terms of this one characterizes categories equivalently as those algebras of the $j^* j_!$-monad which satisfy a Segal condition in that they are in the image of $i_*$.
[ never mind ]
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