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Conventionally, adjunction spaces are defined for two topological spaces, i.e. $X \cup_f Y$ is the space formed from attaching $Y$ to $X$ along some continuous map $f:A \rightarrow X$, where $A$ is a closed subset of $Y$. Some authors (see e.g. Brown's Topology and Groupoids, pg 125) give a generalisation in which multiple spaces $Y_i$ are attached to the same $X$. However, the generalisation stops here.

My first guess at the next generalisation up would be to remove the space $X$ and instead have some sort of family $Y_i$ of topological spaces, subspaces $A_{ij}$ and continuous maps $f_{ij}$ such that each $Y_i$ and $Y_j$ form a standard binary adjunction space under the map $f_{ij}: A_{ij} \rightarrow Y_j$. This would mean the $Y_i$ are all stuck to each other, instead of to a single space $X$. Perhaps particular conditions on the subspaces $A_{i j}$ of $Y_i$ and the functions $f_{i j}$ need to be imposed in order to ensure that this space is well-defined, but I'm sure this is possible.

Is there some canonical resource for generalisations of adjunction spaces in this direction? Or better yet, is there some well-known resource for everything adjunction spaces?

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    $\begingroup$ The original construction is just a pushout. More generally you can consider other colimits. $\endgroup$ – Qiaochu Yuan Sep 28 '18 at 18:53
  • $\begingroup$ @QiaochuYuan ok, I don't know much category theory. Can you suggest a resource that covers generalised pushouts? $\endgroup$ – Doc Sep 29 '18 at 10:52

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