Products in the slice category Sets/I

Question

I'm trying to work my way through a category theory textbook on my own (Awodey's) and came across a problem asking about groups in a slice category $\mathbf{Sets}/I$ for any set $I$. For a group to exist in a category, Awodey said it must have finite products, which makes sense, since we need products to define a binary operation. Which led me to think about what a product is in $\mathbf{Sets}/I$

I figured the obvious candidate would likely be an arrow whose domain is the product in $\mathbf{Sets}$ but that doesn't seem to work, as follows.

Let $I$ be the set $\{1,2\}$, then consider the sets $\{1\}$, $\{2\}$. For objects in the slice category, take functions $i,j$ which will be the embeddings $\{1\}\rightarrow \{1,2\}$ and $\{2\}\rightarrow \{1,2\}$ respectively. Suppose there is a product $i\times j$ which has as its domain the set $\{(1,2)\}$. Let $\bar 1,\bar 2:\{(1,2)\}\rightarrow \{1,2\}$ denote the constant functions which map to 1 and 2 respectively.

Then, $i\circ \bar 1:\{(1,2)\}\rightarrow \{1,2\}$ is the constant function which maps to 1, and $i\circ\bar 2$ is the constant function which maps to 2. There's no single choice for $i\times j$ which makes it an actual candidate to be a product. (I also don't think disjoint union works, it doesn't satisfy the UMP I don't think.)

So then, what is a product in $\mathbf{Sets}/I$, and if it is some function from the product in $\mathbf{Sets}$ where's the mistake in my reasoning above?

Answer 1

The product in a slice category is the fiber product or pullback. Thinking of the slice category over a set $I$ as the category of $I$-indexed sets, it is the "pointwise" product. In your example, the product is empty.

More explicitly, if $f : X \to I$ and $g : Y \to I$ are objects in the slice category of sets over $I$, then their product is

$$X \times_I Y = \{ (x, y) : f(x) = g(y) \}$$

where the function to $I$ is the common value of $f$ and $g$. More evocatively, writing $X_i = f^{-1}(i), i \in I$ and similarly for $Y$, we have

$$(X \times_I Y)_i = X_i \times Y_i.$$

As a side comment, it is in fact not necessary for a category to have products to define group objects in it. Here is a definition which does not require the existence of products and which reduces to the usual definition if products exist:

A group object in a category $C$ is a functor $C^{op} \to \text{Grp}$ whose underlying functor to $\text{Set}$ is representable.

Basically the point is that you can instead take products in presheaves over $C$, which always exist, and which agree with products in $C$ under the Yoneda embedding if those exist. For example, by this definition the cyclic group $C_2$ of order $2$ is meaningfully a group object in the category of sets of size at most $2$, even though the product $C_2 \times C_2$ doesn't exist.