Does the category of Simplicial Complexes have finite limits and colimits?

Question

Does the category of Simplicial Complexes have finite limits and colimits?

Does the geometric realization functor preserve them?

Thanks!

Answer 1

Let $\mathbf{S}$ be the category of simplicial complexes in the sense you described. Consider the full subcategory spanned by the $n$-simplices $\Delta^n$. It is not hard to see that this is isomorphic to the full subcategory $\mathbf{F}$ of $\mathbf{Set}$ spanned by the sets $[n] = \{ 0, \ldots, n \}$, so we get a functor $N : \mathbf{S} \to [\mathbf{F}^\mathrm{op}, \mathbf{Set}]$ by sending a simplicial complex $Y$ to the presheaf $\mathbf{S}(\Delta^{\bullet}, Y)$. For brevity, we write $X_n$ instead of $X([n])$ when $X$ is an object in $[\mathbf{F}^\mathrm{op}, \mathbf{Set}]$.

Proposition. The functor $N : \mathbf{S} \to [\mathbf{F}^\mathrm{op}, \mathbf{Set}]$ so defined is fully faithful and has a left adjoint.

Proof. It is clear that $N$ is faithful, because a morphism of simplicial complexes is determined by its action on vertices. It is not hard to check that it is also full. Let $\mathrm{cosk}_0 : \mathbf{Set} \to [\mathbf{F}^\mathrm{op}, \mathbf{Set}]$ be the functor that sends a set $V$ to the presheaf $\mathop{\mathrm{cosk}_0} V$ defined by $(\mathop{\mathrm{cosk}_0} V)_n = \mathbf{Set}([n], V)$. Clearly, every object $X$ in $[\mathbf{F}^\mathrm{op}, \mathbf{Set}]$ admits a unique morphism $X \to \mathop{\mathrm{cosk}_0} X_0$ that acts as the evident bijection $X_0 \to \mathbf{Set}([0], X_0)$ in degree 0. Define $L X$ to be the image of $X$ in $\mathop{\mathrm{cosk}_0} X_0$. Then, the hom-set map $$[\mathbf{F}^\mathrm{op}, \mathbf{Set}](L X, N Y) \cong [\mathbf{F}^\mathrm{op}, \mathbf{Set}](X, N Y)$$ induced by the canonical morphism $X \to L X$ is a bijection. I claim that $L X$ is isomorphic to $N K X$ for some simplicial complex $K X$. Indeed, take the vertex set of $K X$ to be $X_0$, and declare $A \subseteq X_0$ to be an $n$-simplex if $A$ is the image in degree 0 of some monomorphism $\Delta^n \to L X$. Since $N$ is fully faithful, the naturality of the above bijection then implies that $K$ is a functor $[\mathbf{F}^\mathrm{op}, \mathbf{Set}] \to \mathbf{S}$ and is the required left adjoint for $N$. 　◼

Corollary. $\mathbf{S}$ has limits and colimits for all small diagrams.

Proof. This is a standard fact about reflective subcategories: it is easy to check that the colimit of a diagram $Y : \mathcal{J} \to \mathbf{S}$ can be computed as $L({\varinjlim}_\mathcal{J} N Y)$, and that $N : \mathbf{S} \to [\mathbf{F}^\mathrm{op}, \mathbf{Set}]$ creates all (small) limits.　◼

It is not clear to me whether or not the geometric realisation functor $\mathbf{S} \to \mathbf{Top}$ preserves colimits. However, it does not preserve even binary products: in $\mathbf{S}$, $\Delta^n \times \Delta^m \cong \Delta^{n m + n + m}$ (because the Yoneda embedding $\mathbf{F} \to [\mathbf{F}^\mathrm{op}, \mathbf{Set}]$ preserves products, and $N \Delta^n$ is isomorphic to the presheaf represented by $[n]$), which has the wrong dimension! This is one reason to prefer simplicial sets over simplicial complexes.