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Stream: theory: category theory

Topic: slice functor factorization

Bruno Gavranović (Oct 14 2022 at 19:36):

It’s known that the covariant slice functor $\mathcal{C} \to \mathbf{Cat}$ (on arrows defined by postcomposition) can be factored through the Yoneda embedding $y : \mathcal{C} \to [\mathcal{C}^{op}, \mathbf{Set}]$ and the category of elements $\Sigma: [\mathcal{C}^{op}, \mathbf{Set}] \to \mathbf{Cat}$. Is it known whether there’s an analogous factorization for the contravariant slice construction which is on arrows defined by pullback?

El Mehdi Cherradi (Oct 18 2022 at 09:00):

Well, any functor to a cocomplete category factors (up to isomorphism) through the Yoneda embedding followed by its left Kan extension along the latter (which is the factorization of your example), but I don't think that is what you are looking for ?

Bruno Gavranović (Oct 20 2022 at 07:21):

I didn't know about that. Do you have a reference/some more elaboration for this?

Trying to understand what you said - If a functor to a cocomplete category factors through a Kan extension along the Yoneda embedding, then it means the natural transformation in the Kan extension isn't just a natural transformation, but an isomorphism, right?

El Mehdi Cherradi (Oct 20 2022 at 08:40):

This nlab page discusses this particular case : https://ncatlab.org/nlab/show/free+cocompletion (Proposition 2.1).
It is a very general result that taking a (normal or even homotopy) Kan extension along a fully faithful functor then restricting along the same functor gets you back where you started from up to isomorphism; you can have a look at the Examples section here for instance : https://ncatlab.org/nlab/show/exact+square#fully_faithful_functors.

dusko (Oct 22 2022 at 08:56):

Bruno Gavranovic said:

It’s known that the covariant slice functor $\mathcal{C} \to \mathbf{Cat}$ (on arrows defined by postcomposition) can be factored through the Yoneda embedding $y : \mathcal{C} \to [\mathcal{C}^{op}, \mathbf{Set}]$ and the category of elements $\Sigma: [\mathcal{C}^{op}, \mathbf{Set}] \to \mathbf{Cat}$. Is it known whether there’s an analogous factorization for the contravariant slice construction which is on arrows defined by pullback?

if you perform the category of elements construction before you begin, and look at $[\mathcal{C}^{op}, \mathbf{Set}] \simeq \check{\cal C}$ as discrete fibrations over $\cal C$ and $[\mathcal{C}, \mathbf{Set}]^{op}\simeq \hat{\cal C}$ as discrete opfibrations, then there is the Isbell adjunction $y^*\dashv y_*: \hat{\cal C} \to \check{\cal C}$, where $y^*$ takes any discrete fibration to the discrete opfibration of cocones out of it as a diagram, whereas $y_*$ takes any discrete opfibration to the discrete fibration of cones under it. now note your covariant slice functor the yoneda embedding into discrete fibrations followed by the domain. the contravariant slice should be going int discrete opfibrations, whose domains are the coslice categories. the contravariant embedding into slices and pullbacks is not a strict functor, but a pseudofunctor, and it might be better to collect it and look at the corresponding fibration. otherwise, if you count on being able to choose the pullback functors between the slice categories coherently, then choosing the coherent isomorphisms eg with respect to any subgroups of $\cal C$ will require that you split group epimorphisms, which sometimes works and sometimes does not.