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Stream: theory: category theory

Topic: discrete objects

Daniel Teixeira (Sep 09 2024 at 18:10):

Is there a name for categories $\mathcal V$ with a functor (embedding?) $\mathsf{Set}\to \mathcal V$ which admits both left and right adjoints? Or concrete category whose forgetful functor $U:\mathcal V\to\mathsf{Set}$ has a left adjoint which has a left adjoint. The idea is to define objects in $\mathcal V$ which are discrete

fosco (Sep 09 2024 at 18:20):

That seems exactly how they are defined! https://ncatlab.org/nlab/show/discrete+object#definition

fosco (Sep 09 2024 at 18:23):

An alternative approach (a clever trick) to define a discrete object in a 2-category is to say that an object $D$ of $\mathbf K$ is discrete if for every object $A$ the hom-category $\mathbf K(A,D)$ is discrete. The idea comes from here and recovers the fact that functors into a discrete category are just functions on objects between the underlying sets of objects, in the 2-category of small categories

Kevin Carlson (Sep 09 2024 at 18:34):

The definition on nLab looks a bit different to me: it says discrete objects exist when $U$'s left adjoint is fully faithful, which roughly says that a discrete object doesn't contain any new stuff that wasn't already in its underlying set. Topological spaces have discrete objects in this sense, but annoyingly the discrete space functor doesn't have a further left adjoint in general, basically because a product of discrete topological spaces need not be discrete. If this nonexistent left adjoint did exist then it would be the connected components functor, which is the starting point of axiomatic cohesion.

Mike Shulman (Sep 10 2024 at 05:45):

If $\mathcal{V}$ is a topos, then these adjunctions make it a [[locally connected topos]].