Questions tagged [internal-categories]

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Let $C$ be a category with pullbacks. Recall that a colimit in $C$ is van Kampen if it is preserved by $C/(-): C^{op} \to CAT$, $c \mapsto C/c$. We can $C$-internalize everything in sight: Let $\... 0answers 180 views Internal$2$-categories Has the notion of an internal$2$-category been studied, or more generally an internal$n$-category? Do we have any examples of naturally occurring internal$2$-categories/$n$-categories? The ... 0answers 116 views Cartesian liftings in double categories The question: I wonder whether the following definition, or something similar, has appeared somewhere (see below for motivations). Any reference or pointer is welcome! (In what follows, I denote ... 0answers 156 views Maximal algebraic sub-groupoids By a theorem of Ehresmann, topological and Lie categories (by which I mean categories internal to$Top$and$Diff$respectively, with the condition that the source and target in the latter case are ... 0answers 88 views 'The' object of composable triples in an internal category In any category$\mathcal{C}$with pullbacks, we can define an internal category$\mathscr{C}$in$\mathcal{C}$as an object${\bf Ob}_\mathscr{C}$of objects and an object${\bf Hom}_\mathscr{C}$of ... 0answers 86 views Pushforward of an internal category along a functor Let$F:C\to D$be a “nice” functor (for example,$H_*(-;\mathbb{Z}):\mathbf{Top}\to \mathbf{Ab}^{\mathbb{Z}}$). Now assume that we have a category$O$internal to$C$. Is there a canonical way to ... 0answers 93 views Reclusive Categories Has there been any work done on internal categories inside internal categories? I'm familiar with$n$-fold categories, but I don't want an internal category inside the category of internal categories ... 0answers 108 views Colimits of n-fold categories An$n$-fold category is an internal category in the category of$(n-1)$-fold categories (and a$0$-fold category is just a Set). General results about internal categories assure that the category of$...
Note: Expanded and rephrased, per Todd's question below. Suppose that we have a set-valued functor $S:\mathcal{C}\to\mathbf{Sets}$, and an arrow $p:Y\to X$ such that $S(p)$ has finite fibers. From ...