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Let ${\bf C}$ be a category, and $\zeta$ an associative binary operation on a set $I$. Define a $5$-tuple $(D_o,D_h,u,v,d)$ as follows: $D_o$ is just the object class of $\bf C$, while $D_h$ is the quotient of $\coprod_{i \in I} \hom({\bf C})$ by the equivalence $\mathcal R$ that amalgamates two pairs $(f,i)$ and $(g,j)$ iff $f=g={\rm id}_{\bf C}(A)$ for some $A \in {\rm Ob}({\bf C})$; $u$ and $v$ are the functions $D_h \to D_o$ taking the equivalence class $(f,i) \bmod \mathcal R$ to $s(f)$ and $t(f)$, respectively; and $d$ is the mapping $D_h \times_{v,u} D_h \to D_h$ sending a pair $((f,i)\bmod \mathcal R,(g,j) \bmod \mathcal R)$ to $(f,i) \bmod \mathcal R$, respectively to $(g,j) \bmod \mathcal R$, if $g$, respectively $f$, is an identity of $\bf C$, and to $(fg,\zeta(i,j)) \bmod \mathcal R$ otherwise.

It is clear that $(D_o,D_h,u,v,d)$ is a category, let me denote it by $\zeta \ast \bf C$. Since I don't know if $\zeta \ast \bf C$ has already a name, I'm calling it the augmentation of $\bf C$ by $\zeta$: While looking forward to finding something more "natural", I'm using it as a toy example in reference to the idea of a generalized notion of limit which, on the one hand, can make sense in more abstract settings than categories (e.g., Mitchell's semicategories, or even Ehresmann's neocategories [1]), and on the other hand, may perhaps be an effective surrogate of limits in situations where limits do not exist and neither weak limits nor sublimits (see note 2) are likely to represent, for some reason, a satisfactory alternative. So, here are my questions:

Q1. Is $\zeta \ast \bf C$, up to equivalence, a disguised form/a special instance of anything familiar to you? Q2. Does it satisfy any "obvious" universal property? Q3. Is it maybe a particular colimit (see note 2)?

Thanks in advance, as always, for any hint.

Notes. (1) Sublimits are the same as weak limits, save for the fact that we replace "at least" with "at most". (2) Say, as an object of ${\bf Cat}(\mathcal U)$, for a sufficiently large universe $\mathcal U$.

Bibliography.

[1] A. Bastiani and C. Ehresmann, Categories of sketched structures, Cahiers Topologie Géom. Différ. Catégoriques, Vol. 13, No. 2 (1972), 104-214.

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  • $\begingroup$ I am likely confused about some aspect of your construction, but I don't see how $\zeta \ast C$ can be a category unless $\zeta: \alpha\times\alpha \to \alpha$ is associative. If $\zeta$ is associative, then this is (almost, but not quite) the operation of "base change" of a category: if $C$ is any category, and $(\alpha,\zeta,a\in\alpha)$ any associative algebra, then "$C \otimes \alpha$" is the category with the same objects as $C$ and morphisms $\hom_{C\otimes \alpha}(x,y) = \hom_C(x,y)\times \alpha$. I have no useful comments about colimits. $\endgroup$
    – Theo Johnson-Freyd
    Commented Apr 7, 2013 at 3:05
  • $\begingroup$ Thanks, Theo, for your comment. Yes, sorry, I forgot to say that $\zeta$ must be associative, in the categorial case. Let me edit and fix it. $\endgroup$
    – Salvo Tringali
    Commented Apr 7, 2013 at 9:48
  • $\begingroup$ Even if $\zeta$ is associative, the composition in $\zeta * C$ is typically not associative, so it is not a category. Suppose that $f$ is an isomorphism in $C$ which is not an identity, and let $g$ be an arrow in $C$ with the same codomain. Then consider the two ways to compose $(f,i)\circ (f^{-1},j) \circ (g,k)$. If you do the left composition first, you get $(g,k)$. If you do the right composition first, you get $(g,ijk)$ where I'm writing the operation of $\zeta$ as concatenation. Typically $ijk \neq k$ so these do not agree. $\endgroup$
    – Tim Campion
    Commented Jan 6, 2015 at 18:00

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