# Reference for (Generalised) Adjunction spaces

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?

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