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Stream: learning: questions

Topic: Model category structures on Span(C)

Sergei Burkin (Apr 08 2024 at 23:00):

Let category $C$ be either the category of sets, finite sets or (finitely generated or all) $k$-vector spaces. Let $Span(C)$ be the corresponding 1-category of spans, with composition given essentially by pullbacks, but made to be strictly associative.

1. Do these categories have all equalizers? I cannot find a counter-example.

If it helps, the category $Span(FinSet)$ is equivalent to the category of matrices with entries in $\mathbb{N}$. Equalizer of $f, g:A\to B$ would correspond to a convex cone in $A$ closed under taking sum with rational coefficients (as long as the value of the sum is still in $A$, not just in $A\otimes \mathbb{Q}$). Is the size of Hilbert basis of such a convex cone less or equal to the dimension of $A$?

2. In case these categories do not have equalizers, is there still any point in trying to find possible model category structures on these categories?

El Mehdi Cherradi (Apr 09 2024 at 11:43):

1. There is a counterexample in 4. here (spelled out for $\mathbf{Rel}$, but it also works for $\mathbf{Span}$).
2. There are notions of "nice" relative categories that need not be complete/cocomplete, such as fibration categories. I don't know if it could be the case in your example though.