I have spelled out in the detail the example (now here) of the evaluation map on simplicial function complexes which are nerves of inertia groupoids.

]]>added pointer to:

- Charles Rezk, Section 15 of:
*Introduction to quasicategories*(2022) [pdf]

added description (here) of the evaluation map on the function complex

]]>Looking over the entry now, I have adjusted the previous notation a little in order to avoid a couple of clashes:

the notation “Hom” for the simplicial hom-complex I have replaced with “$Hom_\Delta$”

the letter “$C$” used to denote both the ambient category as well as a simplicial set tensor factor. I have changed to writing “$\mathcal{C}$” for the former and “$S$” for the latter.

Also expanded the presentation just a little.

]]>added pointer to:

- Paul Goerss, J. F. Jardine, Section I.5 of:
*Simplicial homotopy theory*, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) [doi:10.1007/978-3-0346-0189-4, webpage]

Added:

Suppose $C$ is a category that admits small coproducts.

Given simplicial objects $A,B\in C^{\Delta^{op}}$,
their **function complex** is a simplicial set

whose set of $n$-simplices is the set of maps

$\Delta^n\otimes A\to B,$where $\otimes$ denotes the copowering of simplicial objects over simplicial sets given by

$(C\otimes D)_n=\coprod_{i\in C_n}D_n.$The original definition of a function complex in the generality stated above is due to Daniel M. Kan:

- Daniel M. Kan,
*On c.s.s. categories*, Boletín de la Sociedad Matemática Mexicana 2 (1957), 82–94. PDF.