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Let $\mathscr C$ be a category. Suppose for any collection $\{c_i\}_{i\in \mathscr C_1}$ ($\mathscr C_1$ denotes the collection of morphisms in $\mathscr C$), the product $\prod_{i\in \mathscr C_1} c_i$ exists in $\mathscr C$. Then $\mathscr{C}$ is a preorder: $\forall a,b \in \mathscr C$, $Hom_{\mathscr C}(a,b)$ has at most one element.

Proof: Suppose $\mathscr{C}$ is not a preorder, then there exists $a,b\in \mathscr{C}_0$($\mathscr C_0$ denotes the collection of morphisms in $\mathscr C$) and $f,g: a\to b$ with $f\neq g$.

Consider a constant function $j$ $:\mathscr{C_1}\to \mathscr{C_0}$ by $j(h)=b, \forall h\in \mathscr {C_1}$. Let $(\prod_{i\in \mathscr C_1}b,\{p_j: \prod_{\mathscr C_1 }b\to b\}_{j\in \mathscr{C_1}})$ be the corresponding product. By the universal property of products, we have a bijection: $Hom_{\mathscr C} (a,\prod_{i\in \mathscr C_1}b)\to \prod_{i\in \mathscr C_1 }Hom_{\mathscr C}(a,b). $

$|\prod_{i\in \mathscr C_1 }Hom_{\mathscr C}(a,b)|\geq 2^{|\mathscr {C_1}|}$ and $|Hom_{\mathscr C} (a,\prod_{i\in \mathscr C_1}b)|\leq |\mathscr{C_1}|$ (contradiction)

My question:

$1:$ how to understand the set $\{c_i\}_{i\in \mathscr C_1}$? What is the form of the objects in this set? Since the index is a collection of morphisms, it confused me. (Also for the product $\prod_{i\in \mathscr C_1} c_i$)

$2:$ How can we conclude $|\prod_{i\in \mathscr C_1 }Hom_{\mathscr C}(a,b)|\geq 2^{|\mathscr {C_1}|}$ ? What I know is that given a set $X$, there is a bijection between $\mathscr P(X)$ and $2^{|X|}$, where $2=\{0,1\}$.

I am new to the category theory. Much appreciated to any help!

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1 Answer 1

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  1. It's maybe easier to think about $\{c_i\}_{i\in\mathscr{C}_1}$ as a choice, namely for each morphism $f$ in $\mathscr{C}$ you have chosen a particular object $c_f$ in $\mathscr{C}$. Note that $c_f$ and $f$ do not need to be related in any way. The set $\{c_i\}_{i\in\mathscr{C}_1}$ is then the set whose elements are those chosen $c_f$, where $f$ ranges over all morphisms in $\mathscr{C}$. Likewise, the product $c:=\prod_{i\in\mathscr{C}_1}c_i$ is an object of $\mathscr{C}$ that comes with projection maps $\pi_f\colon c\to c_f$ for each morphism $f$ in $\mathscr{C}$, and satisfies the universal property of the product: a map $d\to c$ in $\mathscr{C}$ is the same datum as a collection of maps $d\to c_f$ for each morphism $f$ in $\mathscr{C}$.
  2. The universal property of the product is exactly saying that $\mathrm{Hom}(a,\prod_{i\in\mathscr{C}_1}b)$ is in bijection with $\prod_{i\in\mathscr{C}_i}\mathrm{Hom}(a,b)$ (namely, we use the above description of $\prod_{i\in\mathscr{C}_1}c_i$ in the special case where $c_f=b$ for any morphism $f$ in $\mathscr{C}$). As such, $$|\mathrm{Hom}(a,\prod_{i\in\mathscr{C}_1}b)|=|\prod_{i\in\mathscr{C}_i}\mathrm{Hom}(a,b)|=\prod_{i\in\mathscr{C}_i}|\mathrm{Hom}(a,b)|=|\mathrm{Hom}(a,b)|^{|\mathscr{C}_1|}.$$ Since we assumed $|\mathrm{Hom}(a,b)|\geq 2$, this yields $|\mathrm{Hom}(a,\prod_{i\in\mathscr{C}_1}b)|\geq 2^{|\mathrm{Hom}(a,b)|}$.
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