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Stream: learning: questions

Topic: spans for chosen class of pullbacks

Cole Comfort (Oct 15 2020 at 11:55):

Given a category $\mathbb X$, the category of "formal spans" in $\mathbb X$ can be defined to have morphisms $X_1\to X_n$ given by a composable sequence of spans.

$X_1 \leftarrow A_1 \rightarrow X_2 \cdots X_{n-1}\leftarrow A_{n-1} \rightarrow X_n$

Where composition is just formally pasting composable spans together.

If $\mathbb X$ has pullbacks, the 1-category of spans in $\mathbb X$ can be obtained by quotienting these so called "formal spans" by pullback. Suppose that instead of pulling back all cospans, one only has a class of chosen pullbacks in $\mathbb X$ to quotient by. Is there a name/reference for such a construction?