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A pointed category is a (small) category together with a distinguished object, called the basepoint $*$. How does one explicitly construct the coproduct of two pointed categories?

This problem reduces to constructing the coproduct of two pointed connected categories. If both categories have one object, this becomes the free product of monoids.

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Well it is the pushout of $C \leftarrow 1 \rightarrow D$. Colimits of categories can be constructed by "merging" generators and relations, as it is the case with every algebraic category. [Of course, here $2$-colimits are pretty natural, but they turn out to be far more complicated!] Explicitly, $C \cup_1 D$ is the following category: It has objects $\mathrm{ob}(C) \cup_{\star} \mathrm{ob}(D)$, where we identify the two base objects $\star$. A morphism from, say, $c \in C$ to $d \in D$ is a finite chain of morphisms

$c \to c' \to \star \to d \to d' \to \star \to c'' \to c''' \to \star \to \dotsc \to d$.

We have the obvious cancellation rules. When $C,D$ have just one object, namely $\star$, this is the usual construction of the "free product" (this terminology is quite misleading since it is not a product, but rather a coproduct) of monoids.

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  • $\begingroup$ Do all small colimits in ${\textbf{Cat}}$ exist? $\endgroup$ – Gao 2Man Apr 8 '12 at 14:39
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    $\begingroup$ Yes, because it is algebraic. A category consists of two sets $O,M$ with two maps $M \to O$ (source, target), a map $O \to M$ (identity) and $M \times_O M \to M$ (composition), such that certain equalities hold. $\endgroup$ – Martin Brandenburg Apr 8 '12 at 15:29
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It is useful to consider this construction in more generality, as is well written up in the case of groupoids (see Higgins' book "Categories and groupoids"). Let $C$ be a small category, and $f: Ob(C) \to Y$ a function. Then there is a category $f_*(C)$ with object set $Y$ and satisfying a universal property with regard to functors on $C$ whose object function factors through $f$. If $C,D$ are two categories with base points, then you get the free product by taking the disjoint union of $C$ and $D$ and then identifying the base points.

To put this in a more general light, the objects of a (small) category can be regarded as a functor Ob from the category $\mathsf{Cat}$ of small categories and functors to the category $\mathsf{Set}$ of sets and functions, and then this functor Ob is both a fibration (this is about pullbacks) and cofibration (this is about pushouts). There is a basic account of these ideas, which originated with Grothendieck, in

R. Brown and R. Sivera, `Algebraic colimit calculations in homotopy theory using fibred and cofibred categories',  Theory and Applications of Categories, 22 (2009) 222-251.

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  • $\begingroup$ I should have added that the construction $f_*$ is relevant to topology in the groupoid case, for example in investigating the change in the fundamental groupoid $\pi_1(X,A)$ on a set of base points under some identifications on $A$; there need to be some conditions of course, but they apply if say $X$ is a CW-complex and $A$ a set of vertices of $X$. $\endgroup$ – Ronnie Brown Apr 9 '12 at 13:47

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