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Stream: learning: questions

Topic: Concise description of the presheaf model of MLTT?

Fernando Chu (Oct 10 2025 at 07:30):

I'm learning about the presheaf model of MLTT following this note by Coquand, and I believe they are defining the following CwF given a category $\textbf{C}$:

• Category of contexts: presheaves on $\textbf{C}$.
• Presheaf of types: $Ty:\hat{\textbf{C}}^\text{op} \xrightarrow{\int} \textbf{Cat}^\text{op} \xrightarrow{\textbf{Cat}[-^\text{op},\textbf{Set}]} \textbf{Set}$
• Context extension: For a presheaf $\Gamma$ and a type $A:(\int\Gamma)^\text{op}\to \textbf{Set}$, the presheaf $\Gamma.A$ is given by $\Gamma.A(X):=\sum_{\gamma:\Gamma(X)}A(X,\gamma)$.

Is there a nicer way to describe this structure? For example, like how the groupoid model can be seen as the CwF induced by the universal isofibration $\pi:\int (\textbf{1}_{\textbf{Grpd}}) \to \textbf{Grpd}$. I'd also appreciate any insight on this structure, e.g. how to describe context extension or universes in a point free way.

Tom de Jong (Oct 10 2025 at 08:18):

Maybe this note by Thomas Streicher is of interest? https://www2.mathematik.tu-darmstadt.de/~streicher/FIBR/natmod.pdf

Fernando Chu (Oct 10 2025 at 08:29):

It is indeed of interest and quite helpful, thanks!