What on Earth is a homotopy pushout of $$A \rightarrow B \leftarrow C \ \ \ \ \ ???$$ Here $A,B,C$ are elements of a category ${\mathcal V}$ enriched in topological spaces (any convenient category of topological spaces will do). I understand that it is some kind of a weighted limit. This means that I need to take all objects of my category $X$, consider all pushout diagrams $${\mathcal V}(X,A) \rightarrow {\mathcal V}(X,B) \leftarrow {\mathcal V}(X,C)$$ and take their homotopy pushouts $${\mathcal V}(X,A) \times^{h}_{{\mathcal V}(X,B)} {\mathcal V}(X,C) \; .$$ Now the homotopy pushout is just an object $D$ representing this functor, up to weak homotopy equivalence: $${\mathcal V}(X,D) \ \ \ \stackrel{h}{\sim} \ \ \ {\mathcal V}(X,A) \times^{h}_{{\mathcal V}(X,B)} {\mathcal V}(X,C) \; .$$ This is all grand and dandy but how under the Moon can I get my teeth into this object? Are there any explicit ways of calculating it, similar to the methods, working for the bog down homotopy pushout?

Please, do not try to put any model category structure on ${\mathcal V}$. I am semi-clear how to proceed in the case when one can do it.

What on Earth is a homotopy pullback of $$A \rightarrow B \leftarrow C \ \ \ \ \ ???$$ Here $A,B,C$ are elements of a category ${\mathcal V}$ enriched in topological spaces (any convenient category of topological spaces will do). I understand that it is some kind of a weighted limit. This means that I need to take all objects of my category $X$, consider all pullback diagrams $${\mathcal V}(X,A) \rightarrow {\mathcal V}(X,B) \leftarrow {\mathcal V}(X,C)$$ and take their homotopy pullbacks $${\mathcal V}(X,A) \times^{h}_{{\mathcal V}(X,B)} {\mathcal V}(X,C) \; .$$ Now the homotopy pullback is just an object $D$ representing this functor, up to weak homotopy equivalence: $${\mathcal V}(X,D) \ \ \ \stackrel{h}{\sim} \ \ \ {\mathcal V}(X,A) \times^{h}_{{\mathcal V}(X,B)} {\mathcal V}(X,C) \; .$$ This is all grand and dandy but how under the Moon can I get my teeth into this object? Are there any explicit ways of calculating it, similar to the methods, working for the bog down homotopy pullbacks and pushout?

Please, do not try to put any model category structure on ${\mathcal V}$. I am semi-clear how to proceed in the case when one can do it.

What on Earth is a homotopy pushout of $$A \rightarrow B \leftarrow C \ \ \ \ \ ???$$ Here $A,B,C$ are elements of a category ${\mathcal V}$ enriched in topological spaces (any convenient category of topological spaces will do). I understand that it is some kind of a weighted limit. This means that I need to take all objects of my category $X$, consider all pushout diagrams $${\mathcal V}(X,A) \rightarrow {\mathcal V}(X,B) \leftarrow {\mathcal V}(X,C)$$ and take their homotopy pushouts $${\mathcal V}(X,A) \times^{h}_{{\mathcal V}(X,B)} {\mathcal V}(X,C) \; .$$ Now the homotopy pushout is just an object $D$ representing this functor: $${\mathcal V}(X,D) \ \cong \ {\mathcal V}(X,A) \times^{h}_{{\mathcal V}(X,B)} {\mathcal V}(X,C) \; .$$ This is all grand and dandy but how under the Moon can I get my teeth into this object? Are there any explicit ways of calculating it, similar to the methods, working for the bog down homotopy pushout?

Please, do not try to put any model category structure on ${\mathcal V}$. I am semi-clear how to proceed in this case when one can do it.

What on Earth is a homotopy pushout of $$A \rightarrow B \leftarrow C \ \ \ \ \ ???$$ Here $A,B,C$ are elements of a category ${\mathcal V}$ enriched in topological spaces (any convenient category of topological spaces will do). I understand that it is some kind of a weighted limit. This means that I need to take all objects of my category $X$, consider all pushout diagrams $${\mathcal V}(X,A) \rightarrow {\mathcal V}(X,B) \leftarrow {\mathcal V}(X,C)$$ and take their homotopy pushouts $${\mathcal V}(X,A) \times^{h}_{{\mathcal V}(X,B)} {\mathcal V}(X,C) \; .$$ Now the homotopy pushout is just an object $D$ representing this functor, up to weak homotopy equivalence: $${\mathcal V}(X,D) \ \ \ \stackrel{h}{\sim} \ \ \ {\mathcal V}(X,A) \times^{h}_{{\mathcal V}(X,B)} {\mathcal V}(X,C) \; .$$ This is all grand and dandy but how under the Moon can I get my teeth into this object? Are there any explicit ways of calculating it, similar to the methods, working for the bog down homotopy pushout?

Please, do not try to put any model category structure on ${\mathcal V}$. I am semi-clear how to proceed in the case when one can do it.