# Writing categories as slice categories

Is there a known method to write any category $C$ as being equivalent to a slice category bundle $\bar{C}_{/c}\to\bar{C}$, where $C\simeq \bar{C}_{/c}$? It seems one can try to find a category $\bar{C}$ and some $c\in\bar{C}$ with $\int \bar{C}[-,c] \to \bar{C}$ giving the appropriate bundle, where $\int$ denotes the category of elements. Is this a reasonable approach? Where can I read more about this problem? And, is there some kind of free construction that produces a bundle with the property I am looking for?

The motivation for my question is to find out when the correspondence taking objects in $C$ to slice bundles over $C$ can be reversed, so that I can transport problems about functors between slice bundles down to problems about objects in the base category.

Edit for clarity. I am looking for what characterizes slice categories without the context of their domain projection. What intrinsic properties distinguish slice categories? As Todd pointed out, they always have terminal objects. Their colimits are reflected by their projection functors, so I imagine we don't have to worry about many colimits existing since the unsliced category may not have many colimits to begin with.

• No, certainly not, since slice categories have a terminal object. But if $C$ does have a terminal object $1$, then trivially $C \simeq C/1$. – Todd Trimble Sep 23 '16 at 17:21
• Yes, I was thinking about that, but can't we freely give a category a terminal object if it does not have one? – Mathemologist Sep 23 '16 at 17:24
• Of course, but then you're changing the category $C$ (to something inequivalent to $C$). Not sure what you're driving at... – Todd Trimble Sep 23 '16 at 17:25
• Yes, I suppose I need to weaken what I am looking for. In the case that $C$ has a terminal object, what other obstructions do we have? I think what I am trying to do is classify what categories are equivalent to slice categories. – Mathemologist Sep 23 '16 at 17:28
• But my first comment told you: a category is equivalent to a slice category if and only if it has a terminal object. – Todd Trimble Sep 23 '16 at 18:00

The slice category $C/c$ is the category of element of the representable presheaf $h_c \colon C \to \bf Set$. There is a universal construction characterizing this category, which is being a weighted colimit (these thingssatisfy universal props similar to "conical" co/limits). The nLab page about the category of elements contains all the relevant infos!