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This is basically a restatement of this question.

Two arrows $f,g$ are orthogonal, i.e satisfy $f\perp g$, iff the square below is a pullback

$$\require{AMScd} \begin{CD} \mathsf C(B,X) @>{f^\ast}>> \mathsf C(A,X)\\ @V{g_\ast}VV @VV{g_\ast}V\\ \mathsf C(B,Y) @>>{f^\ast}> \mathsf C(A,Y). \end{CD}$$

Suppose now $\mathsf C$ is closed and we replace the hom-sets by internal homs. $$\require{AMScd} \begin{CD} [B,X] @>{[f,-]}>> [A,X]\\ @V{[-,g]}VV @VV{[-,g]}V\\ [B,Y] @>>{[f,-]}> [A,Y]. \end{CD}$$ Is this square still a pullback iff $f\perp g$? What if $\mathsf C$ is a topos (not well-pointed in general)?

I ask because definition 8.10 of these SDG notes by Kostecki seemingly define formal étaleness by using such an internal analogue of orthogonality.

Also, in the case $\mathsf C$ is a topos, can anything be said about the merely existence or uniqueness (without existence) of diagonal fillers in terms of the induced map $[B,X]\to [A,X]\times _{[A,Y]}[B,Y]$ being respectively epic or monic? This would shed light on formal unramifiedness/smoothness for me.

Added. A similar definition of orthogonality (unique lifting) is 5.1 here.

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  • $\begingroup$ As a rule of thumb, when passing from "external" to "internal" it is likely that the internal version will be the parameterized external version, where you have to figure out what "parametrized" means. $\endgroup$
    – Andrej Bauer
    Commented Aug 28, 2016 at 11:30

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The second square being a pullback is a strictly stronger condition than $f\perp g$. Mapping out of the unit object shows that it implies $f\perp g$; mapping out of other objects says that the strong condition is equivalent to $(W\otimes f) \perp g$ for all objects $W$ (or $W\times f$ in the cartesian-closed case; I'm not sure if you meant to restrict to that one).

By mapping out of the unit object, we see that if the induced map $[B,X] \to [A,X]\times_{[A,Y]} [B,Y]$ is monic, then fillers are unique if they exist. And if this map is split epic, or more generally if it belongs to a class of maps with respect to which the unit object is projective, then fillers always exist. But in general in a non-well-pointed topos this map could be epic without any filler existing globally; epimorphy of this map says only that fillers exist "locally" in the sense determined by the topology of the topos.

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  • $\begingroup$ First of all thank you very much for the answer. Where can I read about the details of "fillers exist "locally" in the sense determined by the topology of the topos"? Regarding the first paragraph - it looks like the well-pointedness of the topos of sets is what makes both conditions equivalent for the unenriched case. But for the enriched case, algebraic geometers usually take the definition in terms of the weaker conditions of lifting properties. Why then does SDG take the stronger ones? $\endgroup$
    – Arrow
    Commented Aug 28, 2016 at 13:14
  • $\begingroup$ (not just SDG - the linked definition 5.1 too. Why take this stronger condition as a definition?) $\endgroup$
    – Arrow
    Commented Aug 28, 2016 at 13:42
  • $\begingroup$ I don't know of anywhere where "local existence of fillers" is written down. But we know what an epimorphism is in a sheaf topos (a map with local sections), so just apply that definition to the morphism in question. $\endgroup$
    – Mike Shulman
    Commented Aug 29, 2016 at 8:03
  • $\begingroup$ As for the why to choose one definition or another, I'm sure there are specific concrete reasons as well, but in general the point of SDG is to do mathematics "in a world where everything is smooth", which means in the internal logic of the topos in question. So there is no "external" weaker notion of lifting property; the internal one is all you can state. (Put differently, in the internal logic of a topos, that topos "looks well-pointed".) $\endgroup$
    – Mike Shulman
    Commented Aug 29, 2016 at 8:05
  • $\begingroup$ Thanks again for the comments. It looks like the weaker notion is still possible to state - just using the usual definition of orthogonality, i.e without addressing pullbacks of hom-objects and just asking for fillers. Isn't this a reasonable alternative in SDG too? $\endgroup$
    – Arrow
    Commented Aug 29, 2016 at 8:40

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