Consider category $\mathcal{C}/Z$ consists of objects over Z, i.e arrows over $Z$. Let $h: H \rightarrow Z$ be a fixed object in $\mathcal{C}/Z$. Let $F=Hom(H,X)$. Show that under $F$, pullbacks over $Z$ are just products in the category of sets.
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The usual notation for $\mathcal{C}Z$ is $\mathcal{C}/Z$, and it is called the slice category. I assume $Mor_Z$ is intended to be the hom-functor of $C_Z$; you should use less ambiguous notation. In that vein, the objects are arrows, so it should be $Mor{\mathcal{C}/Z}(h,-)$. Is the $X'$ an accident? Is it supposed to be just $X$? What categories does $F$ go between? (Obviously the codomain is $\mathbf{Set}$.) – Derek Elkins left SE Sep 03 '17 at 10:42
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@DerekElkins Thank you. I editted the question. I was intending $X'$ to be a different object but it doesn't seem to matter. Also, $Mor_Z$ denotes morphisms over $Z$ (I obtained this notation from Lang's Algebra). – S_j Sep 03 '17 at 18:43
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What does an element of $Mor_Z(H,X)$ look like? – Derek Elkins left SE Sep 03 '17 at 20:23
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@DerekElkins $Mor_Z(H,X)$ are morphisms in the category of objects over $Z$. So an element of $Mor_Z(H,X)$ would be a commutative triangle $H \rightarrow X$ (where both $H, X$ have maps to $Z$) such that the composite $H \rightarrow X \rightarrow Z$ is equal to the map $H \rightarrow Z$. – S_j Sep 04 '17 at 01:56
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Here is a possible direction to solve the exercise:
- prove that fiber products of morphisms with common codomain $Z$ are products in the category $\mathcal C/Z$
- (Enter the Yoneda) use the fact that representable copresheaves preserve limit to conclude the proof.
If you need additional hints feel free to ask.
Addedum(since the OP asked :) ): Your functor $F$ is nothing but the representable functor associated to the object $h \colon H \to Z$, hence by general results it preserves product: that is it sends product diagrams in $\mathcal C/Z$ into product diagrams in $\mathbf{Set}$.
By (1) products in $\mathcal C/Z$ are fiber products in $\mathcal C$, hence putting all together we get that $F$ sends the fiber products of $\mathcal C$ (i.e. the products of $\mathcal C/Z$) into products of $\mathbf{Set}$.
Giorgio Mossa
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Isn't 1. is straightforward from definition of product and slice category ? – S_j Sep 04 '17 at 07:40
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@S_j I guess that's depend on you. Everything can be trivial or incredibly complex depending on the person who's considering the problem. – Giorgio Mossa Sep 05 '17 at 06:13
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I mean, saying that something is straightforward is not actually a proof, it's a trick to avoid giving details of a proof. – Giorgio Mossa Sep 05 '17 at 06:14
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would you mind explain a bit further why would step 2 conclude the proof ? (I am very new to category theory) – S_j Sep 06 '17 at 05:37
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@S_j I made a little addendum, take a look and let me know if that helps. – Giorgio Mossa Sep 06 '17 at 08:50
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