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The category $Cat$ of small categories is complete. Could you spell out with details the construction of the limit of a functor $F : J \to Cat$ by products and equalizers? (Mac Lane, Categories for the Working Mathematician, Chapter V, Section 2, Theorem 2)

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  • $\begingroup$ Note that the construction of a limit of a functor by means of products and equalizers works in any category (in which these equalizers and products exist), not just $Cat$. $\endgroup$
    – Abel
    Apr 12, 2016 at 18:23
  • $\begingroup$ @Abel: I want to instantiate this construction on $Cat$. $\endgroup$
    – Bob
    Apr 12, 2016 at 18:28

2 Answers 2

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The limit of a functor $F :B \to C$ can be constructed as the equalizer of

$$s,t :\prod_A FA \longrightarrow \prod_{f : A \to B}FB$$

where $s$ and $t$ are the unique morphisms defined by $$ \pi_{(f : A \to B)}s =\pi_B\\ \pi_{(f : A \to B)}t = (Ff)\pi_A $$ and the universal cone is given by composing with the projections of $\displaystyle\prod\limits_A FA$.

In case $C = Cat$, note that $FA$ is a category for all $A \in B$, and $s$, $t$, and $Ff$ are functors. The functor $s$ is defined on an object $(X_A)_A \in \displaystyle\prod\limits_A FA$ as

$$s((X_A)_A) = (X_B)_{f : A \to B}$$

that is, the $(f : A \to B)$-component of $s((X_A)_A)$ is $X_B$. The functor $t$ is defined on $(X_A)_A$ as

$$t((X_A)_A) = ((Ff)(X_A))_{f : A \to B}$$

On the morphisms of the product category $\displaystyle\prod\limits_A FA$ (these are families of morphisms $g_A : X_A \to Y_A$ in $FA$ for each $A \in B$), the functors $s$ and $t$ are defined by the same formulas:

$$s((g_A)_A) = (g_B)_{f:A\to B}$$

and

$$t((g_A)_A) = ((Ff)(g_A))_{f:A\to B}$$

Finally, note that the limit appears as the subcategory of $\displaystyle\prod\limits_A FA$ consisting of objects and morphisms equalized by $s$ and $t$. Explicitly, an object of the limit $L$ is $(X_A)_A \in \displaystyle\prod\limits_A FA$ such that for all $f : A \to B$ we have $X_B = (Ff)(X_A)$, and a morphism between objects $(X_A)_A$ and $(Y_A)_A$ in $L$ is a family of morphisms $(g_A : X_A \to Y_A)_A$ in $\displaystyle\prod\limits_A FA$ such that $g_B = (Ff)(g_A)$ for all $f : A \to B$.

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  • $\begingroup$ This is indeed exactly what is stated in Mac Lane's book. $\endgroup$
    – Bob
    Apr 12, 2016 at 18:34
  • $\begingroup$ How do you define (not simply specify) $s$ and $t$ in the case of $Cat$? $\endgroup$
    – Bob
    Apr 12, 2016 at 18:42
  • $\begingroup$ @Bob I've added explicit definitions of the functors $s$ and $t$. These simply follow from the construction of the product in $Cat$ and its universal property. $\endgroup$
    – Abel
    Apr 12, 2016 at 19:33
  • $\begingroup$ @Bob To complete my answer, I've added an explicit description of the limit in $Cat$. $\endgroup$
    – Abel
    Apr 12, 2016 at 19:38
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Limits of categories are given by limits of their underlying graphs (forget the composition operation,) which are easy to compute when you observe that a graph is just a set-valued functor from the category $\bullet\stackrel{\to}{\to}\bullet$. Sorry for ugly typesetting.

That is, the limit of a diagram of small categories $(C_i,i\in I)$ is just the corresponding limit of the sets of morphisms $(C_i^1)$ and the sets of objects $(C_i^0)$, with source and target operations induced from those of the $C_i$. The identities are composition are naturally determined by those in the $C_i$, so if you can compute limits of sets, you can compute limits of categories. N.B. This doesn't work at all for colimits-many colimits of categories are heinous! Categories are monadic over graphs, but monadic functors don't generally preserve colimits.

To hew closer to MacLane's plan and to illustrate this, let's compute products and equalizers. Products are very straightforward: objects of $\prod C_i$ are tuples $(c_i)_{i\in I}$ and morphisms tuples $(f_i)_{i\in I}$ with $c_i$ and objects and $f_i:c_i\to c'_i$ a morphism of $C_i$. The composition is componentwise and the identity of $(c_i)$ is $(\text{id}_{c_i})$ the family of identities of the objects $c_i$. Similarly, equalizers: the equalizer of $F,G:C\to D$ has objects $c$ of $C$ such that $F(c)=G(c)$ and morphisms $f:c\to c'$ such that $F(f)=G(f)$, with identities and composition inherited from $C$. As advertised, in both cases we just take the limit of morphism sets and the limit of object sets, and connect them in the natural way. And this explicit description allows you to use the general theorem to give an explicit description of arbitrary limits of categories-which we've seen will reduce to just using the explicit description of arbitrary limits of sets, twice.

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    $\begingroup$ A related answer to your first part is that the category of small categories is equivalent to the category of models of the essentially algebraic theory of categories which is equivalent to the category of finite limit preserving functors into $\mathbf{Set}$ from a particular, easy to describe, category. This functor category has all limits. The end result is similar. $\endgroup$ Apr 11, 2016 at 23:29
  • $\begingroup$ I can spell out the objects of the limit category but I have a hard time spelling out its morphisms. Can you help? $\endgroup$
    – Bob
    Apr 12, 2016 at 12:54
  • $\begingroup$ Are you having trouble with the products or equalizers? That's all you really need, right? $\endgroup$ Apr 12, 2016 at 16:24
  • $\begingroup$ No problem with products and equalizers. But then how do I combine them to get limits? $\endgroup$
    – Bob
    Apr 12, 2016 at 17:27
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    $\begingroup$ This was in fact the point of my question, if you read it again. $\endgroup$
    – Bob
    Apr 12, 2016 at 17:54

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