hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
homotopy theory, (∞,1)-category theory, homotopy type theory
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see also algebraic topology
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The simplicial mapping complex or simplicial hom complex of two simplicial set is the collection of maps, left homotopies and higher homotopies between them, all itself organized into a simplicial set.
More formally, SimplicialSets is a Cartesian monoidal category and the corresponding internal hom is what is traditionally known as the simplicial mapping complex.
Since SimplicialSets is the category of presheaves over the simplex category, its internal hom has the general form discussed at closed monoidal structure on presheaves:
For $X, Y \,\in\, sSet$, their internal hom or simplicial mapping complex is the simplicial set
whose
component set in degree $k$ is the hom-set of simplicial sets
from the product of simplicial sets of $X$ with the standard n-simplex $\Delta[n]$
to $Y$,
and whose face maps $d_i$ and degeneracy maps $s_i$ are given by precomposition with $id_X \times \delta_i$ and $id_X \times \sigma_i$, respectively ($\delta_i$ and $\sigma_i$ denoting the generating morphisms of the simplex category).
(simplicial mapping complex between nerves of groupoids) For $N(\mathcal{G}_i)$ the nerves of groupoids, their simplicial mapping complex is isomorphic to the nerve of the functor groupoid from $\mathcal{G}_1$ to $\mathcal{G}_2$:
Last revised on July 12, 2021 at 04:09:32. See the history of this page for a list of all contributions to it.