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I am interested in the category-theoretic description of trees (and operads?) and have started a course of study that will allow me to engage with these two (1, 2) manuscripts of Joachim Kock.

An essential prerequisite to the early portions of the manuscript involves an understanding of a pair of adjoint functors. The left adjoint is the change of basis functor $g^{\ast}: \mathbf{Set}_{/A} \rightarrow \mathbf{Set}_{/B}$ associated with the change of basis between sets $g: B \rightarrow A$. I find it very easy to reason about the image of a bundle under the change of basis functor. This functor takes a bundle $f: X \rightarrow A$ to the pullback of $f$ by $g$, which I can easily interpret in $\mathbf{Set}_{/B}$ as the fibered product with the canonical projection $h: X \times_{A} B \rightarrow B$ given by $h: (x, b) \mapsto b$.

Kock utilizes the right adjoint, which I have read is called the dependent product: $$ (g^* \dashv \prod_g) \colon \mathbf{Set}_{/B} \stackrel{\overset{g^* }{\leftarrow}}{\underset{\prod_g}{\to}} \mathbf{Set}_{/A} \,. $$

I am trying to understand this functor, but am finding it very difficult. Are there any set-theoretic descriptions of the image of a bundle $f: X \rightarrow B$ under $\prod_g$ that would help me in this context?

Although I've only had a little exposure to it, the process of re-describing adjoint functors in the "internal language of their categories" (I hesitate to use this phrase here-- I have seen it used in the literature of toposes and do not know its formal definition-- perhaps I'll just call this process reifying?) has proven very difficult for me. Are there any mental algorithms that offer any help? Are there a set of useful exercises I can undertake to develop the skill? Are there theorems I can study that will provide insight into the process?

It has struck me in the course of mulling over the problem that I would also like to know whether category-theorists even bother with reifying their constructions. Does this process have a name? Is it done often?

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    $\begingroup$ A sequence of exercises that may give insight into the process of "reification": 1. For an object $X$ in $\operatorname{Set}/A$ and an element $a\in A$, show that the preimage of $a$ may be identified with $\operatorname{Hom}_{\operatorname{Set}/A}{a,X}.$ 2. Use this description and the definition of right adjoint to compute the fibers of the dependent product. $\endgroup$ – dhy Jan 15 at 6:52
  • $\begingroup$ It may be helpful to explain what your background is. For me, coming from an algebraic geometry/sheaf-theory background, I look at this and I see that you're asking for the right adjoint of a pullback, so the answer should look like a pushforward. But different fields talk about similar notions in different language, and so the most helpful answer may be from someone who speaks your language. (BTW, most but not all category-theorists do have in mind explicit models of their constructions.) $\endgroup$ – dhy Jan 15 at 6:56
  • $\begingroup$ @dhy it's a question about sets, so one can write down a fairly concrete description. $\endgroup$ – David Roberts Jan 15 at 6:59
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    $\begingroup$ That's fair; the point I was trying to make was that every field has a list of "standard adjunctions", and that in practice these sorts of questions are often answered by comparing to those "standard adjunctions." But I see that my comment could easily be interpreted as "this way (via pushforwards & pullbacks) is the best way to think about it." $\endgroup$ – dhy Jan 15 at 7:07
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    $\begingroup$ Hint: A good way to bootstrap this is first to consider the case where A=✱ is the set with a single element. The general case can then be treated by applying the special case to both A and B. Edit: As soon as I left this comment, David gave an answer that does exactly this! =]. $\endgroup$ – Harry Gindi Jan 15 at 7:13
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It's worth thinking about the simplest example, namely when $A$ is a single point. In this case, $g^*$ is the product functor $-\times B$, and its right adjoint is the set of sections: it sends $f\colon X\to B$ to the set $\Gamma(B,X)$ of sections $s\colon B\to X$ of $f$. This is vaguely similar behaviour to how the internal hom in a cartesian closed category is right adjoint to taking a product. In particular, if $X = Y\times B = g^*Y$, then $\Pi_g f = Y^B$.

The space of sections is the same thing as the product of all the fibres of $f$, so one can think of the general dependent product this way, now just parameterised by the elements of $A$. Given a point $a\colon *\to A$, which is an object of $\mathbf{Set}/A$, we can calculate $$ \mathbf{Set}/A(a,\Pi_g f) \simeq \mathbf{Set}/B(B\times_A *,f) = \mathbf{Set}/B(g^{-1}(a),f) $$ Now the latter is the set $\Gamma(g^{-1}(a),X)$ of sections of $f$ over $g^{-1}(a)\subset B$. But $\mathbf{Set}/A(a,\Pi_g f)$ is the fibre of $\Pi_g f \to A$ over $a$, so $$ \Pi_g f = \coprod_{a\in A} \Gamma(g^{-1}(a),X) \to A. $$

A similar mental picture works, to some extent, in categories that behave like the category of sets, except that you shouldn't think of $\Pi_g f$ as a disjoint union, but something more like a continuous family of spaces of sections. Note howoever, how I used the defining property that it was a right adjoint to get a handle on the construction. In general, one can use generalised elements in place of $1\to A$ to construct the thing, at least as a presheaf, and then knowing the dependent product exists means this is representable.

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