General categorical definitions always have two variants, a strict one, in which associativity and unity hold as equalities, and a weak one, in which they hold up to equivalence. However, every definition I've seen for higher categories assumes that the source of a composite $f; g$ is equal to the source of $f$ (in diagrammatic notation) and the target of $f; g$ is equal to that of $g$. Is there a deeper reason for this? I can see that in the usual ways to define weak higher categories, such as simplices or weak enrichment, these source-target rules are fulfilled strictly, but theoretically this could be a weakness of the models, and I could for instance conceive of a cubical setting in which they are not. Is there some theorem stating that categorical structures defined with sources and targets of composites only equivalent to those of their components are always equivalent to ones in which they are defined to hold on the nose, or is there another reason to assume so?
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$\begingroup$ If you have morphisms $f:A\to B$, $g:C\to D$ and an equivalence $\alpha: B \overset{\sim}{\to} C$, then you can can always take $g \circ \alpha \circ f$ as the "weak composition" of $f$ and $g$. $\endgroup$– Antoine LabelleCommented Oct 14, 2023 at 14:42
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$\begingroup$ @AntoineLabelle That weakens the typing of which morphisms can be composed. What I mean is, if we weaken the axioms of a categorical structure, wouldn't we also need to weaken the source-target rule by only providing equivalences $α: s(f;g) \simeq s(f)$ and $β: t(f;g)\simeq t(g)$, along with maybe some coherence axioms? Are you saying that one can use a weak composition as you describe to transition from such a structure with weak source-target rules to one with strict ones? $\endgroup$– Alexander PraehauserCommented Oct 14, 2023 at 17:18
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$\begingroup$ Wouldn't this be circular? When you define the composition of morphisms by using a sort of equivalence which in turn is defined by two inverse morphisms meaning that their compositions are equal or homotopic to the identity - where to start? $\endgroup$– Martin BrandenburgCommented Oct 14, 2023 at 22:11
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$\begingroup$ @MartinBrandenburg This circularity is inherent in weak higher category theory, which is why we use structures like homotopy types or geometric constructions to circumvent it. But consider for instance a cubical setup, in which objects are points and morphisms lines. One can define composition with weak source-target rules by giving for each pair of lines $f, g$ a lift of these lines to a square, having one side given by the path $f,g$, where the composite $f;g$ is defined as the line opposite to the path $f,g$. In this setup, the square acts as an equivalence. $\endgroup$– Alexander PraehauserCommented Oct 15, 2023 at 9:17
2 Answers
However, every definition I've seen for higher categories assumes that the source of a composite f;g is equal to the source of f (in diagrammatic notation) and the target of f;g is equal to that of g.
This is not true for many models of higher categories.
For example, (∞,1)-categories can be modeled by simplicial spaces (i.e., bisimplicial sets), interpreted as objects in the model category of simplicial spaces equipped with Rezk's model structure for complete Segal spaces.
Composition is defined using the zigzag $$X_1 ⨯^h_{X_0} X_1 ← X_2 → X_1,$$ where the left map is a weak equivalence whenever $X$ is a local object, i.e., satisfies the Segal conditions and the completeness condition. A point in the left space is a pair of morphisms $g:Y_1→Z$, $f:X→Y_0$, together with a path (i.e., a zigzag of 1-simplices) $h$ from $Y_0$ to $Y_1$.
To define the composition $g∘f$, we lift the triple $(f,g,h)$ through the left map, obtaining a point $p∈X_2$, unique up to a weakly contractible choice, and then apply the right map, obtaining a point $q∈X_1$.
The source and target of $q$ are homotopic to the source of $f$ and the target of $g$, but there is no reason they have to be equal.
If we force $X$ to be Reedy fibrant, then it is possible to ensure that the source and target of $q$ are equal to the source of $f$ and the target of $g$. However, there are many examples of simplicial spaces that are not Reedy fibrant, e.g., some of the most natural definitions of bordism categories tend to produce simplicial spaces (or $n$-fold simplicial spaces) that are not Reedy fibrant.
Dmitri Pavlov’s answer points out models where these equations are indeed weakened. But at the same time, there are good reason why they’re strict in many models of weak higher categories: These equalities have a very different character from the associativity/unitality equations. In a type-theoretic terminology, they are part of the type of composition/units. Saying the same thing homotopy-theoretically, they assert that some newly-posited map into a fibration agrees with a pre-existing map into the base. In some models these statements are literally true; in others, they’re at least motivation, and (sometimes) can be made into coherence theorems.
Saying the latter version precisely: Suppose we’re setting up higher categories internal to a fibration category $\mathcal{E}$ (or some similar setting). We assume the category is given with a fibrant object of 0-cells $C_0$, and a fibration of 1-cells $(s,t) : C_1 \to C_0 \times C_0$ (and objects of higher cells similarly fibered over their boundaries in some way). Then composition is is a map $C_1 \times_{C_0} C_1 \to C_1$, over the map $(s \pi_0, t \pi_1) : C_1 \times_{C_0} C_1 \to C_0 \times C_0$. In other words, it’s a lift of $(s \pi_0, t \pi_1)$ into the fibration of 1-cells. Similarly, the “units” map is a lift of the diagonal $C_0 \to C_0 \times C_0$ into the fibration of 1-cells.
So we can see this as specified in a framework where each component of our structure is either:
- a new fibration over some given base (constructed from things already assumed);
- or, given a map $f : X \to Y$ and a fibration $p : Z \to Y$ (constructed from earlier data), a new map $g : X \to Z$ that is a lift of $f$ along $p$.
(But note we don’t allow ourselves to posit an arbitrary equality between existing maps, nor that an existing map is a fibration.)
Then this framework allows defining models of higher categories with source/target laws for all compositions holding on the nose, but does not allow imposing strict associativity or unitality constraints. There are substantial bodies of work developing this framework and closely-related ones (Reedy diagrams, FOLDS-structures, (cofibrant) generalised algebraic theories, etc), and their models in homotopical settings, and showing how they are homotopically better-behaved than finite-limit theories involving arbitrary equations.