# Notation for replacing a substructure by another substructure?

In set theory, suppose $$A\subset B$$. Then suppose we want to "replace $$A$$ by $$C$$". We have a notation for this: $$B'=(B\setminus A) \cup C$$.

However, suppose that $$A,B,C$$ are groups, or topological spaces, and $$A$$ is a subgroup or subspace of $$B$$, and that we want to generate a new group or topological space $$B'$$ by replacing $$A$$ by $$C$$. In this case, in order to remain faithful to the structure, we have to specify a particular way in which it's replaced, so we need extra data. Suppose we have a morphism $$\phi:C\to A$$ (homomorphism or continuous function), which tells us how the replacement is supposed to happen.

Is there a standard notation for denoting the equivalent of replacement $$B'=(B\setminus A) \cup C$$, in general categories other than Set? (I am not sure actually which assumptions about the category would be needed to define this).

Edit: I can imagine that something like $$B[\phi:A\setminus C]$$ is standard notation?

• What do you mean by "replacing"? what purpose does it serve? – Jackozee Hakkiuz Aug 1 at 13:09
• Before discussing notation, you should first define your construction in detail. Think about the topological space case first – that should be comparatively straightforward. After that you might think about the group case. When you have a construction that works in a few well known categories, come back and ask your question again. – Zhen Lin Aug 1 at 13:10

You may be looking for Heyting arrow. It sometimes happens that the poset of subobjects of a fixed object $$A$$ is also a Heyting algebra. This means that it's a distributive lattice and that for every pair of subobjects $$B$$ and $$C$$, there is a subobject denoted $$B\implies C$$ or $$B\succ C$$ such that $$(D\wedge B) \leq C \hspace{5mm} \text{if and only if} \hspace{5mm} D\leq(B\succ C).$$
In the case of subsets of a set $$A$$, then $$\wedge$$ is intersection, $$\leq$$ is containment and I claim that $$B\succ C = B^c\cup C$$, where $$B^c$$ is the absolute complement (i.e., with respect to $$A$$). You can check this: if $$D\cap B\subseteq C$$, then $$D\subseteq C$$ and hence $$D\subseteq B^c\cup C$$. Conversely, if $$D\subseteq B^c\cup C$$, then $$D\cap B \subseteq (B^c\cup C)\cap B=C$$.
This also happens if you are considering the family of open subsets of a topological space. In this case you have $$B\succ C = \text{Int}(B^c\cup C)$$.