I’m studying the category theory with the help of “Abstract and Concrete Categories” book by Jiri Adamek and others.

There I encountered with the notion of functor-structured categories Spa$($T$)$ and I'm trying to find more information about it, because, as I understood, many other categories can be viewed as full subcategories of Spa$(T)$. Recall from definition 5.40 of the book that

For a functor $T: \mathbf{X} \to \mathbf{Set}$, the category $\mathbf{Spa}\,(T)$ has pairs $(X,\alpha)$ as objects where $X \in \mathbf{X}$ and $\alpha \subseteq T(X)$, and arrows $(X,\alpha) \to (Y,\beta)$ are the morphisms $f: X \to Y$ such $T(f)(\alpha) \subseteq \beta$.

So if you know helpful textbooks on this categories to better understand it, tell about them.
Thanks for any help!

  • 1
    You'd probably get more and timelier responses if you inlined the definitions instead of relying upon readers to find the book and then find the definition within the book. For that particular book, my vague recollections of the parts of that book that I read is that its terminology and notations were idiosyncratic. – Derek Elkins Oct 8 '17 at 14:54
up vote 1 down vote accepted

I doubt you find any textbook on such a specific notion.

But you can view this construction as a particular case of a Grothendieck construction. The latter applies to any (pseudo) functor $F : \mathbf X \to \mathbf{Cat}$ and yields a category denoted $\int F$ whose

  • objects are the pairs $(X,a)$ where $X$ is an object of $\mathbf X$ and $a$ an object of $F(X)$,
  • morphisms $(X,a) \to (Y,b)$ are pairs $(f,k)$ where $f: X \to Y$ is a morphism of $\mathbf X$ and $k : F(f)(a) \to b$ is a morphisms of $F(Y)$,
  • composition $(g,\ell)\circ (f,k)$ for $(g,\ell) :(Y,b) \to (Z,c)$ and $(f,k):(X,a) \to (Y,b)$ is given by: $$ X \overset f \to Y \overset g \to Z ,\qquad F(gf)(a) \simeq F(g)(F(f)(a)) \overset {F(g)(k)} \longrightarrow F(g)(b) \overset{\ell} \to c$$

Now if we denote $\operatorname{Sub} : \mathbf{Set}\to \mathbf{Cat}$ the functor that maps each set to the poset of its subsets (viewed as a category), then each functor $T:\mathbf X \to \mathbf{Set}$ induces $$ \dot T : \mathbf X \overset T \to \mathbf{Set} \overset{\operatorname{Sub}} \to \mathbf{Cat}$$ The construction $\mathbf{Spa}\,(T)$ is exactly the Grothendieck construction $\int{\dot T}$.

Keywords to find more on this kind of constructions are: Grothendieck (op)fibration, Grothendieck bifibrations, Lawvere's (hyper)doctrines.

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