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The presheaf on a category $C$ is the category of functors $[C^\mathrm{op}, \mathrm{Set} ]$. For my purposes the presheaf is a free construction to obtain limits/colimits, the free cocompletion of a category.

Can you "defunctionalise" the free cocompletion of a category? By defunctionalize I mean not use so many exponential objects and use more primitive objects instead in the same way a lot of other free constructions have more explicit and less abstract representations?

My thoughts

The "purpose" for presheafs as a free construction is to give you the limits and colimits of diagrams.

$\mathrm{lim}: [D^\mathrm{op},C] \rightarrow \mathrm [1, \mathrm{Pshf} (C)]$

A diagram is usually represented as a functor from some category $D^\mathrm{op}$ to $C$. It's kind of like a little subset of a category I guess.

Could you also represent a diagram as an explicit inductively defined graph? I also feel like you could represent a big graph of morphisms as a functor from out of the category of simplices or some other geometry $[\Delta, C]$.

Am I on the right track or going nowhere? Or does the concept of defunctionalization not make any sense at this level.

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    $\begingroup$ A note on terminology: a "presheaf on $C$" is an object of $[C^{op},\mathrm{Set}]$, not the functor category itself. I don't think I've seen a single instance of $[C^{op},\mathrm{Set}]$ being called a presheaf. $\endgroup$ Commented Apr 9, 2021 at 17:59
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    $\begingroup$ The category of presheaves on $\mathbf C$ is equivalent to the category of discrete fibrations over $\mathbf C$. Is this closer to what you have in mind? $\endgroup$
    – varkor
    Commented Apr 9, 2021 at 18:26
  • $\begingroup$ @varkor I still don't really understand this Grothendieck kind of stuff but it does seem a step closer to what I want. Fib(C) sounds easier to "defunctionalize" than Pshf(C) even if it's not all the way defunctionalized. $\endgroup$ Commented Apr 9, 2021 at 19:24
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    $\begingroup$ I'm not sure that I fully understand your question, but there's another representation of presheaves over category $C$ as extensions of $C$ by a single object $x$ with no endomorphisms (besides $1_x$) and no arrows of the form $x\to c$. This way, we can work with categories and functors in place of presheaf functors and natural transformations. $\endgroup$
    – Berci
    Commented Apr 10, 2021 at 8:35
  • $\begingroup$ @Berci that sounds kind of like a category with an indeterminate arrow but maybe a level up or simpler. Strangely that relates to kurims.kyoto-u.ac.jp/~hassei/papers/ctcs95.html some stuff I've been looking at. And yeah the representation you gave is much more concrete then a functor and probably easier to work with. Iirc you can interpret lambda binders as limits so I shouldn't be surprised. $\endgroup$ Commented Apr 10, 2021 at 17:39

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An explicit description of the free strict small-cocompletion of a locally small category is given in Beurier–Pastor–Guitart's Presentations of clusters and strict free-cocompletions, which I believe is the kind of description you're looking for.

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