# 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 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”.

• As you guess, you want $F$ to preserve pullbacks, which is trouble in this case. – Kevin Carlson Jul 19 at 18:55
• as an example for Kevin's comment (with any coefficients): $S^3 \times_{S^2} S^3 \cong S^3 \times S^1$. – Mike Miller Jul 19 at 20:28
• 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? – FKranhold Jul 19 at 20:42
• @FKranhold What's the enriched definition of a category object? – Denis Nardin Jul 20 at 7:25
• 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”. – FKranhold Jul 20 at 11:32