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Hi there. Suppose ${\bf C}_1$, ${\bf C}_2$ and $\bf D$ are categories and $F_i$ is a functor ${\bf C}_i \to \bf D$. Consider the subcategory of the comma category $( F_1 \downarrow F_2)$ whose objects are all triples $(c_1, c_2, e)$ where $c_i \in \mathrm{obj}(\mathbf C_i)$, $F_1(c_1) = F_2(c_2)$ and $e$ is the identity of $\hom_\mathbf{D}(c)$ with $c := F_i(c_i)$. Is there a standard name for this particular category? Thank you in advance.

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2 Answers 2

up vote 7 down vote accepted

The (strict) pullback of $F_0$ and $F_1$.

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Nothing more specific than this? Ok, I will wait to hear from somebody else, if any, before accepting your answer. Thanks. –  Salvo Tringali Apr 22 '12 at 17:11
3  
I think you can accept Mike's answer now. –  Andrej Bauer Apr 22 '12 at 19:26
    
@Andrej: Great job! –  Martin Brandenburg Apr 22 '12 at 20:48
    
I see, nothing but the (strict) pullback of $F_1$ and $F_2$. Ok! –  Salvo Tringali Apr 22 '12 at 21:08

The (strict) pullback of $F_0$ and $F_1$.

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