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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 23rd 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## 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 $$(C\otimes D)_n=\coprod_{i\in C_n}D_n.$$ ## Related concepts * [[internal hom]] * [[simplicial object]] * [[simplicial set]] ## 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](https://dmitripavlov.org/scans/kan-on-css-categories.pdf). <a href="https://ncatlab.org/nlab/revision/diff/function+complex/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/function+complex/2">v2</a>, <a href="https://ncatlab.org/nlab/show/function+complex">current</a>

   Added:

   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

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

   Related concepts

   • internal hom

   • simplicial object

   • simplicial set

   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.

   diff, v2, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 4th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Paul Goerss]], [[J. F. Jardine]], Section I.5 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkh&#228;user (1999) Modern Birkh&#228;user Classics (2009) &lbrack;[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/function+complex/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/function+complex/3">v3</a>, <a href="https://ncatlab.org/nlab/show/function+complex">current</a>

   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]

   diff, v3, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 4th 2022
   • (edited Dec 4th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexLooking 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. <a href="https://ncatlab.org/nlab/revision/diff/function+complex/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/function+complex/3">v3</a>, <a href="https://ncatlab.org/nlab/show/function+complex">current</a>

   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.

   diff, v3, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 4th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded description ([here](https://ncatlab.org/nlab/show/function+complex#Properties)) of the evaluation map on the function complex <a href="https://ncatlab.org/nlab/revision/diff/function+complex/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/function+complex/3">v3</a>, <a href="https://ncatlab.org/nlab/show/function+complex">current</a>

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

   diff, v3, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 4th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Charles Rezk]], Section 15 of: *Introduction to quasicategories* (2022) &lbrack;[pdf](https://faculty.math.illinois.edu/~rezk/quasicats.pdf)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/function+complex/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/function+complex/4">v4</a>, <a href="https://ncatlab.org/nlab/show/function+complex">current</a>

   added pointer to:

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

   diff, v4, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 5th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have spelled out in the detail the example (now [here](https://ncatlab.org/nlab/show/function+complex#EvaluationMapOnNervesOfInertiaGroupoids)) of the evaluation map on simplicial function complexes which are nerves of inertia groupoids. <a href="https://ncatlab.org/nlab/revision/diff/function+complex/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/function+complex/6">v6</a>, <a href="https://ncatlab.org/nlab/show/function+complex">current</a>

   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.

   diff, v6, current