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I feel one should make a distinction between categories in their so-to-speak metamathematical role, for example in using results such as the Yoneda Lemma, or a left adjoint preserves colimits, and cateories categories used as an algebraic structure with partial operations, in which guise I cite many examples of groupoids.

One such example groupoid is the unit interval groupoid $\mathbf I$ with two objects $0,1$ and exactly one arrow between any objects. This groupoid plays a similar role in the category of groupoids to the integers in the category of groups. Also the integers are obtained from $\mathbf I$ by identifying $0,1$ in the category of groupoids: this is one explanation of why the fundamental group of the circle is the integers.

That last comment is an application of the Seifert-van Kampen Theorem for the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points, a theorem which is about calculating homotopy $1$-types of not necessarily connected spaces. It is quite necessary in using this theorem to keep all the information about the way various components of the pieces intersect; moving to equivalence will destroy that information.

In the case of groupoids with structure, which may be a topology, or a smooth structure, or an algebraic structure (group, ring, Lie algebra,...) then the usual equivalence of a transitive groupoid to a group of course no longer applies to preservation of the extra structure. But there are simpler structures: for example one knows how to classify vector spaces with a single endomorphism, but how does one classify groupoids with a single endomorphism?

Maybe the situation is less clear with categories, rather than groupoids, but it may also be that these two roles, metamathematical, and an algebraic structure with partial operations, merge in some situations. For me, that is one of the fascinations of category (and groupoid) theory.

And my definition of Higher Dimensional Algebra is as the study of partial algebraic structures with operations whose domains are given by geometric conditions.

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I feel one should make a distinction between categories in their so-to-speak metamathematical role, for example in using results such as the Yoneda Lemma, or a left adjoint preserves colimits, and cateories used as an algebraic structure with partial operations, in which guise I cite many examples of groupoids.

One such example groupoid is the unit interval groupoid $\mathbf I$ with two objects $0,1$ and exactly one arrow between any objects. This groupoid plays a similar role in the category of groupoids to the integers in the category of groups. Also the integers are obtained from $\mathbf I$ by identifying $0,1$ in the category of groupoids: this is one explanation of why the fundamental group of the circle is the integers.

That last comment is an application of the Seifert-van Kampen Theorem for the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points, a theorem which is about calculating homotopy $1$-types of not necessarily connected spaces. It is quite necessary in using this theorem to keep all the information about the way various components of the pieces intersect; moving to equivalence will destroy that information.

In the case of groupoids with structure, which may be a topology, or a smooth structure, or an algebraic structure (group, ring, Lie algebra,...) then the usual equivalence of a transitive groupoid to a group of course no longer applies to preservation of the extra structure. But there are simpler structures: for example one knows how to classify vector spaces with a single endomorphism, but how does one classify groupoids with a single endomorphism?

Maybe the situation is less clear with categories, rather than groupoids, but it may also be that these two roles, metamathematical, and an algebraic structure with partial operations, merge in some situations. For me, that is one of the fascinations of category (and groupoid) theory.

And my definition of Higher Dimensional Algebra is as the study of partial algebraic structures with operations whose domains are given by geometric conditions.