Does the category of Simplicial Complexes have finite limits and colimits? Does the category of Simplicial Complexes have finite limits and colimits?
Does the geometric realization functor preserve them?
Thanks!
 A: 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.
