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Stream: theory: category theory

Topic: adjoint of arrow functor

Daniel Plácido (Jul 01 2021 at 16:48):

Is the arrow category functor $\text{Arr}:\mathsf{Cat\to Cat}$ a left adjoint?

Daniel Plácido (Jul 01 2021 at 16:49):

In our context we needed $\text{Arr}$ and $\text{ob}:\mathsf{Cat\to Set}$ to be left adjoints. Turns out the latter is a two-sided adjoint, but the former looks worrying.

Nathanael Arkor (Jul 01 2021 at 17:11):

I think an argument like the following shows that it's not left adjoint. $\mathrm{Arr} = ({-})^\to$, which is right adjoint because Cat is cartesian-closed, but left adjoint only if the interval category $\to$ is tiny. However, Cat is generated under filtered colimits by $\to$, so $\to$ is only finitely presentable, not tiny.

Ivan Di Liberti (Jul 01 2021 at 20:44):

Yes, $\to$ is not tiny. Its associated corepresentable functor $\mathsf{Cat}(\to,-)$ does not preserve, for example, the colimits in the Example 5.3.8 in "Category Theory in Context" by Emily Riehl.

dusko (Jul 02 2021 at 09:53):

Daniel Plácido said:

Is the arrow category functor $\text{Arr}:\mathsf{Cat\to Cat}$ a left adjoint?

you mean taking a category to the category of arrows and commutative squares? it is a left adjoint if you take the codomain to be the category of categories with factorization systems. the canonical factorization in the category of arrows is to factorize each square into two triangles. in other words, the arrow category functor is a monad for factorization systems as algebras.

i think this was in my paper Maps I in JPAA from maybe 93. there was a more complicated way to say it all that was somewhere else, maybe as a side remark, but i don't remember where. could be isbell? the reference should be in the paper.

Daniel Plácido (Jul 05 2021 at 12:54):

thank you all for the inputs, that was exactly the kind of answer I was looking for

Daniel Plácido (Jul 05 2021 at 12:54):

too bad tho!