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Let $\mathcal C$ a category. We define the category of cones $\text{Con}(\mathcal C)$ in the following way:

  • Objects: quadruples $(\mathcal Z, \underline{M}, M, m)$ consisting of a small category $\mathcal Z$, a functor $\underline{M}: \mathcal Z \to \mathcal C$ and a cone $(M, m)$ of $\underline{M}$ (i.e. $M$ is an object in $\mathcal C$ and $m = (m_Z)_{Z \in \mathcal Z}$ is a collection of morphism $m_Z: M \to \underline{M}(Z)$ satisfying a compatibility condition) .
  • Morphisms: a morphism $\underline{\phi}: (\mathcal Z, \underline{M}, M, m) \to (\mathcal Y, \underline{N}, N, n)$ is a triple $\underline{\phi}:( \phi^0, \phi^1, \phi^2)$ where $\phi^0: \mathcal Y \to \mathcal Z$ is a functor, $\phi^1: M \to N$ is a morphism in $\mathcal C$ and $\phi^2: \underline{M}\circ \phi^0 \to \underline{N}$ is a natural transformation such that $\phi^2_Y \circ m_{\phi^0(Y)} = n_Y \circ \phi^1$ for all $Y \in \mathcal Y$.

I am trying to show that the forgetful functor $U: \text{Con}(\mathcal C) \to \mathcal C$ has a left adjoint, i.e. there is a functor $S : \mathcal C \to \text{Con}(\mathcal C)$ such that $$\text{Hom}_{\text{Con}(\mathcal C)}(S(X), Y) \cong \text{Hom}_\mathcal C (X, U(Y)).$$ The more natural way to associate an element $C \in \mathcal C$ to a quadruple in $\text{Con}(\mathcal C)$ is to send $C$ to $(\varnothing, F_\varnothing, C, m)$ where $F_\varnothing$ is the unique functor that goes from $\varnothing$ to $\mathcal C$ and $(C, m)$ is a cone for $F_\varnothing$ (a cone for this functor is just an element, there is no need to consider such an $m$).

This seems to be a natural way to define the functor $S:\mathcal C \to \text{Con}(\mathcal C)$ as I did above but I do not really see how it goes for the morphism. If we consider a morphism $\underline{f} \in \text{Hom}_{\text{Con}(\mathcal C)}(S(C), Y)$ (for $S(C)$ construct above and $Y = (\mathcal Y, \underline{N}, N, n)$ a quadruple in $\text{Con}(\mathcal C)$), $\underline{f}$ should be a triple $\underline{f} = (f^0, f^1, f^2)$ with $f^0: \mathcal Y \to \varnothing$ and $f^3: F_\varnothing \circ f^0 \to \underline{N}$. The problem is that $F_\varnothing \circ f^0 : \mathcal Y \to \mathcal C$ factors trough the empty set and therefore it has no image, $F_\varnothing \circ f^0$ is not even a functor.

My questions are the following: Is it really possible to consider a functor as $f^0$ such that $f^0: \mathcal Y \to \varnothing$ ? Is it the right way to construct the left adjoint $S$ ? I also tried to construct $S$ by sending $C$ to an other quadruble where the associate $\mathcal Z$ is not the empty set but it seems that in those cases there are many possiblities which are all very arbitrary (there is a unique functor $\varnothing \to \mathcal C$ but there are a lots a functor from $\mathcal Z = \underline{1} \to \mathcal C$, where $\underline{1}$ is the category with a unique element, for example).

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  • $\begingroup$ Can you clarify what "$(M,m)$ is a cone of $\underline{M}$" means for you? My best guess is that $M$ is an object in $\mathcal{C}$ and $m$ is a collection of morphisms from $M$ to the objects in the diagram $\underline{M}$ (satisfying the correct compatibility conditions) -- is that right? $\endgroup$ – diracdeltafunk Jan 15 at 19:59
  • $\begingroup$ Yes that's it, is there a more "universal" notation for a cone of a functor ? It is the one we use in category theory course. $\endgroup$ – Falcon Jan 15 at 20:18
  • $\begingroup$ I've edited, hope it is more clear now. $\endgroup$ – Falcon Jan 15 at 20:25
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    $\begingroup$ Why do you define the category of cones like this? It would seem more natural to have the functor go from Z to Y instead (and the rest changed accordingly). $\endgroup$ – Thorgott Jan 15 at 23:49
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    $\begingroup$ Since a morphism $\underline{\phi}: (\mathcal Z, \underline{M}, M, m) \to (\mathcal Y, \underline{N}, N, n)$ involves a functor $\mathcal{Y}\to\mathcal{Z}$, rather than the other direction, and since the left adjoint property of $S(C)$ involves morphisms from $S(C)$, and therefore functors to the category $\mathcal{Z}$ associated with $S(C)$, it makes sense to choose $\mathcal{Z}=\underline{1}$, since that has a unique functor to it from every other category. $\endgroup$ – Jeremy Rickard Jan 16 at 9:58
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Your idea to take the cone over an empty category to represent objects is basically good, however, as was noted in the comments, a morphism from a cone over shape $\mathcal Z$ to a cone over $\mathcal Y$ involves a contravariant part, namely the functor $\phi^0:\mathcal Y\to\mathcal Z$.
This is because the hom functor (of the category of categories) is contravariant in the first argument.

So, instead of the initial category we need to consider the terminal category: the discrete category $\bf 1$ with one object $\ast$.
And, then for an object $X\in\mathcal C$, define $S(X):=({\bf 1},\ \ast\mapsto X,\ X,\ 1_X)$, and verify the claim.

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I've tried to show the statement for $S(C) = (\underline{1}, F_{C}, C, 1_C)$ (where $F_C$ is the functor $\{*\} \to C$) as recommended by Berci and I wanted to know everything is good in the justification, let me know if you guys see anything wrong.

For $\widetilde{Y} = (\mathcal Y, \underline{N}, N, n) \in \text{Con}(\mathcal C)$, we can consider $\underline{\phi} \in \text{Hom}_{\text{Con}(\mathcal C)}(S(C), \widetilde{Y})$ so that $$\underline{\phi} = (\phi^0, \phi^1, \phi^2)$$ where $\phi^0$ is the unique functor that goes from $ \mathcal Y$ to $\underline{1}$, $\phi^1: C \to N$ is any morphism in $\text{Hom}_\mathcal C(C, N) = \text{Hom}_\mathcal C(C, U(\widetilde{Y}))$ and $\phi^2$ is a natural transformation $F_C \circ \phi^0 \to \underline{N}$.

The natural transformation $\phi^2$ requires
$$\phi^2_Y = n_Y \circ \phi^1,$$ since $m_{\phi^0(Y)} = 1_C$. Therefore, $\underline{\phi} \in \text{Hom}_{\text{Con}(\mathcal C)}(S(C), \widetilde{Y})$ is entirely determined by a morphism $\phi^1 \in \text{Hom}_\mathcal C(C, U(\widetilde{Y}))$ since $\phi^0$ is unique and $\phi^2_Y$ is equal to $n_Y \circ \phi^1$. We conclude that $$\text{Hom}_{\text{Con}(\mathcal C)}(S(C), \widetilde{Y})\cong \text{Hom}_\mathcal C(C, U(\widetilde{Y})).$$ (The naturality in $C, \widetilde{Y}$ seems quite ugly so I'm gonna stop here).

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  • $\begingroup$ It's correct. To handle naturality cleanly, there's an approach for adjunctions via profunctors: specifically here we can add so called heteromorphisms $C\to(\mathcal Z,\underline M,M,m)$ to the disjoint union $\mathcal C+{\rm Con}(\mathcal C)$ simply as arrows $C\to M$ (define compositions so that we receive a category), then show that every object in $\mathcal C$ has the reflection $S(C)$ in the subcategory ${\rm Con}(\mathcal C)$ and every cone has its summit as its coreflection in the subcategory $\mathcal C$. $\endgroup$ – Berci Jan 16 at 22:24

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