This question arises from the lecture notes.
The notation $\perp$ means the thing on the left has the left lifting property w.r.t the thing on the right and, at the same time the right has right lifting property w.r.t the left.
For $u:A\to B,v:C\to D$ monomorphisms in $sSet$. And $p:X\to Y$ a morphism in $sSet$. Why do we have such equivalences between the following:
(1) $(A\times D\cup B\times C\hookrightarrow B\times D) \perp (p:X\to Y)$
(2) $(A\to B)\perp (\underline{Hom}(D,X)\to\underline{Hom}(D,Y)\times_{\underline{Hom}(C,Y)}\underline{Hom}(C,X))$
(3) $(C\to D)\perp (\underline{Hom}(B,X)\to\underline{Hom}(B,Y)\times_{\underline{Hom}(A,Y)}\underline{Hom}(A,X))$
Notation:
- $A\times D\cup B\times C$ is defined by $A\times D\cup_{A\times C} B\times C$.\
- $\underline{Hom}(A,Y)_n=Hom_{sSet}(A\times\Delta^n,Y)$\
My progress: I know that there is an adjunction $$Hom_{sSet}(X,\underline{Hom}(Y,Z)\cong Hom_{sSet}(X\times Y,Z)$$ But I struggle to see how it is applied to the colimit and limit such that the diagram is kinda rotated.
Everything is completely elementary, but it takes a long time to dot all the $i$'s: https://arxiv.org/abs/1902.06074