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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 construct a category $F_*O$ which is internal to $D$?

In the case where $C$ and $D$ are monoidal and $O$ is not internal, but only enriched, the crucial property $F$ should fulfill is monoidality, as we need morphisms $Fx\otimes Fy\to F(x\otimes y)$ when defining the composition. Here, it seems that there must be a transformation of pullbacks $Fx\times_{Fz} Fy\to F(x\times_z y)$, but maybe this is unnecessarily complicated.

I would be happy if one can construct such a pushforward for the above example $H_*(-;\mathbb{Z})$. In my case, I had to replace a $\mathbf{Top}$-enriched category $A$ by a $\mathbf{Top}$-internal category $B$ which is “homotopy equivalent” to the original one (when considering $\mathrm{ob}(A)$ as a discrete space), and I want to say something like “$H_*A$ and $H_*B$ are isomorphic as $\mathbf{Ab}^\mathbb{Z}$-enriched categories”.

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    $\begingroup$ As you guess, you want $F$ to preserve pullbacks, which is trouble in this case. $\endgroup$ – Kevin Carlson Jul 19 at 18:55
  • $\begingroup$ as an example for Kevin's comment (with any coefficients): $S^3 \times_{S^2} S^3 \cong S^3 \times S^1$. $\endgroup$ – Mike Miller Jul 19 at 20:28
  • $\begingroup$ Well … it would be enough to have natural morphisms $H_*(X)\times_Z H_*(Y)\to H_*(X\times_Z Y)$. By the way: Maybe the pullback in $\mathbf{Ab}^\mathbb{Z}$ is not the best thing to consider because I want it to generalize the enriched definition where I have maps $H_*(X)\otimes H_*(Y)\to H_*(X\times Y)$. Isn’t there a version of internality with respect to a monoidal structure? $\endgroup$ – FKranhold Jul 19 at 20:42
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    $\begingroup$ @FKranhold What's the enriched definition of a category object? $\endgroup$ – Denis Nardin Jul 20 at 7:25
  • $\begingroup$ By “enriched definition”, I just meant “definition of a category enriched in a monoidal category”. Here, the homology of a $\mathbf{Top}$-enriched category becomes a $\mathbf{Ab}^\mathbb{Z}$-enriched category. It seems that I am looking for a generalization of internal categories such as “enriched in a monoidal category” generalizes “enriched in a category with finite limits”. $\endgroup$ – FKranhold Jul 20 at 11:32

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