A $\Delta$-set is a contravariant functor from the category $\Delta'$ of order-preserving injections to the category of sets (this is essentially what Allen Hatcher calls a $\Delta$-complex).

A main reason for working with simplicial sets instead of $\Delta$-sets should be that they allow quotients (see e.g. Allen Hatcher's nice appendix "CW complexes with simplicial structure" to his Algebraic Topology book: "A major disadvantage of $\Delta$-complexes is that they do not allow quotient constructions"), How does this go well with the fact that the category of functors $\Delta'op\to Sets$ has colimits?

(This question was already asked in a comment on Allen Hatcher's answer to this question on the definition of simplicial complexes. I apologize for asking it twice but there has been no answer given and I am afraid that the reason is - if it's not the silliness of my question - that the comment appears only after pressing the "more comments" button. However, I apologize.)


1 Answer 1


The basic issue is that not every function that we would like to describe between $\Delta$-complexes can be realized by a natural transformation between functors. The lack of degeneracy maps means that no map $X \to Y$ of $\Delta$-complexes that sends any simplex down to a degenerate simplex can be realized by a natural transformation of functors. For example, if $X$ is a $\Delta$-complex interval realizing $[0,1]$ and $Y$ is a $\Delta$-complex realizing $[0,1]^2$, then there is no natural transformation of functors realizing the projection maps $p_i:[0,1]^2 \to [0,1]$.

As a consequence, the category of $\Delta$-complexes does not have enough immediately-available maps between objects to construct the kinds of colimit diagrams one would like to realize.


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