# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 23rd 2022

## Definition

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

$Hom(A,B)$

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

$(C\otimes D)_n=\coprod_{i\in C_n}D_n.$

## References

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.
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeDec 4th 2022

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 4th 2022
• (edited Dec 4th 2022)

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.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeDec 4th 2022

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

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeDec 4th 2022