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I'm looking for references about under-categories under a set of objects. In case this hasn't been studied before, I'll write what I mean by this below. If it has been studied before, but in a different way from what I suggest, then I'd be interested to hear that too. The application I have in mind is for model categories, so if someone can confirm that this theory works for categories then I'll ask about the model category connection in a future question.


Let $C$ be a category $a$ an object. Then one can form $(a \downarrow C)$ by looking at the arrow category $C^J$ for $J = \bullet \to \bullet$, and taking the subcategory where the domain of every arrow is $a$. The morphisms in this category are commutative triangles with $a$ as the top vertex.

Now consider $S = \{a,b\}$ and assume we have maps $a \to b$ and $b\to a$. Then we can form $(S\downarrow C)$ as a subcategory of $C^J$ for $J$ a square, and taking the subcategory where the top of the square is any arrow in {$a\to a$, $a \to b$, $b \to a$, $b\to b$} and the bottom of the square can be any arrow in $C$, as long as the square commutes. Morphisms in this category can be commutative cubes where the entire top face consists of $a$'s and $b$'s, where the front and back faces are elements of ($S\downarrow C$), and where the bottom face can use any elements and morphisms in $C$. This seems like a perfectly good category, but I've never read about it anywhere.

(1) Has this category been studied? Or have I made some error and in fact this isn't a category? Note that I don't mind adding more hypotheses to $S$, if it means there's a theory already developed for this.

Now let $S$ be a set $n$ objects in $C$ and make some hypothesis regarding the existence of maps in $S$ (for instance, that every pair of elements $s_1,s_2$ has morphisms going both ways). Form $(S\downarrow C)$ by taking as objects $n$-cubes (where the top face consists of elements and maps from $S$) and as morphisms $n+1$-cubes as above.

(2) Has this been studied? Can it be done for an infinite set $S$? I'm also interested in the dual question of $(C\downarrow S)$.

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I think that $S\downarrow\mathcal{C}$ (where $S= a\leftrightarrows b$) is different from you description (you need the commutativity respect each morphisms...). Anyway in the book "Mal'cev, Protomodular, Homological and Semi-Abelian Categories" you find a very nice construction about comma categories (see pag. 132 of that book, category of points..), I seem near to your construction (the couple $a, b$ is a splitted couple retraction-section) –  Buschi Sergio Oct 18 '12 at 19:22
    
It looks to me that you'd want to take the (sub)category generated by your two arrows, rather that just the diagram. –  David Roberts Oct 18 '12 at 22:59
    
I've been reading up on comma categories, and checked out Sergio's reference, but I don't see that they are related to the category I constructed above. It seems that in any comma category objects are arrows in $C$, but that is not so for my category. Instead, my objects are squares and my morphisms are cubes. @David Roberts: could you please say a bit more about what you mean? Would what you describe sit inside $Arr(C)$? Would it be the comma category or the one I wrote about? Thanks! –  David White Nov 3 '12 at 18:47
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