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I´m learning model categories. So, I have the following question.

The canonical example of model category, besides the category of topological spaces, is the category $\Delta^{o}Set$ consisting of simplicial sets. One of the requirements for a category to be a model category is that has to be closed under finite inductive limits and projective limits.

Question. Knowing that the category of sets $Set$ all finite limits and colimit exists, why is it true that $\Delta^{o}Set$ is closed under finite inductive limits and projective limits?

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  • $\begingroup$ Terminology nitpick : I think what you mean is not closed under limits/colimits, but has limits/colimits. "Closed under" is used for subcategories of a given category (for example : the category of abelian groups is closed under limits in the category of groups). $\endgroup$ – Arnaud D. Aug 24 '17 at 9:28
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The category of simplicial sets $\mathbf{Set}^{\Delta^\mathrm{op}}$ is an example of a functor category, and if a category $\mathbf{C}$ has all limits/colimits of shape $\mathcal{I}$, then the functor category $\mathbf{C}^\mathbf{D}$ has all limits/colimits of shape $\mathcal{I}$, and moreover such limits/colimits can be computed pointwise. See, e.g., section V.3 in Mac Lane's Categories for the Working Mathematician.

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