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semicategory

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higher category theory

Basic concepts

Basic theorems

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Morphisms

Functors

Universal constructions

Extra properties and structure

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Contents

Idea

The notion of semicategory or non-unital category is like that of category but omitting the requirement of identity-morphisms.

This generalizes the notions of semigroup, semiring, etc:

Semicategories, like categories, appear as semipresheaves on the category with two objects and two morphisms.

Definition

Definition

A (small) semicategory or non-unital category 𝒞\mathcal{C} consists of

  • a set 𝒞 0\mathcal{C}_0 of objects;

  • a set 𝒞 1\mathcal{C}_1 of morphisms (or arrows);

  • two functions s,t:𝒞 1𝒞 0s, t : \mathcal{C}_1 \to \mathcal{C}_0 called source (or domain) and target (or codomain);

    • one writes f:xyf : x \to y if s(f)=xs(f) = x and t(f)=yt(f) = y;
  • a function :𝒞 1× t,s𝒞 1𝒞 1\circ \colon \mathcal{C}_1 \times_{t,s} \mathcal{C}_1 \to \mathcal{C}_1 (composition) from the set of pairs of morphisms such that the target of the first is the source of the second;

such that the following properties are satisfied:

  • source and target are respected by composition: s(gf)=s(f)s(g \circ f) = s(f) and t(gf)=t(g)t(g\circ f) = t(g);

  • composition is associative: (hg)f=h(gf)(h \circ g)\circ f = h\circ (g \circ f) whenever t(f)=s(g)t(f) = s(g) and t(g)=s(h)t(g) = s(h).

Remark

If one added to this definition the existence of a function i:C 0C 1i \colon C_0 \to C_1 such that for all cC 0c \in C_0 the morphism i(c)i(c) is an identity on cc under the given composition, then one has the defintion of a category.

However, having identities is just an extra property on a semi-category, not extra structure. For more on this see below at Relation to categories.

Remark

One often writes hom(x,y)hom(x,y), hom C(x,y)hom_C(x,y), or C(x,y)C(x,y) for the collection of morphisms f:xyf : x \to y; the latter two have the advantage of making clear which category is being discussed. People also often write xCx \in C instead of xC 0x \in C_0 as a short way to indicate that xx is an object of CC. Also, some people write Ob(C)Ob(C) and Mor(C)Mor(C) instead of C 0C_0 and C 1C_1.

Definition

For 𝒞,𝒟\mathcal{C}, \mathcal{D} two semicategories, a semi-functor F:𝒞𝒟F \colon \mathcal{C} \to \mathcal{D} is a pair of functions F 0:𝒞 0𝒟 0F_0 \colon \mathcal{C}_0 \to \mathcal{D}_0, F 1:𝒞 1𝒟 1F_1 \colon \mathcal{C}_1 \to \mathcal{D}_1 that respects all the given structure in the obvious way.

Write SemiCatSemiCat for the (large) category whose objects are semicategories, and whose morphisms are semifunctors.

Properties

Relation to categories

We discuss the relation of semicategories to categories. (See for instance the beginning of (Harpaz) for a quick review of basics, with an eye towards their generalization to the relation between complete Segal spaces and complete semi-Segal spaces.)

Definition

There is an evident forgetful functor

U:CatSemiCat U \colon Cat \to SemiCat

from the category Cat of categories to that of semicategories, def. 2, given simply by forgetting the identity-assigning map i:𝒞 0𝒞 1i \colon \mathcal{C}_0 \to \mathcal{C}_1 in a category.

Definition

For 𝒞\mathcal{C} a semi-category, def. 1, write

Id(𝒞 1)𝒞 1 Id(\mathcal{C}_1) \hookrightarrow \mathcal{C}_1

for the subset on those morphisms which are endomorphisms on some object x𝒞 0x \in \mathcal{C}_0 and such that they are neutral elements in their endomorphisms semimonoids End 𝒞(x)End_{\mathcal{C}}(x).

Proposition

A semicategory is the semicategory underlying a category, hence is in the image of the functor UU of def. 3, precisely if every object has a neutral endomorphism, hence precisely if the composite diagonal function in

Id(𝒞 1) 𝒞 1 s 𝒞 0 \array{ Id(\mathcal{C}_1) &\hookrightarrow& \mathcal{C}_1 \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{s}} \\ && \mathcal{C}_0 }

is an isomorphism, where the horizontal function is that of def. 4.

Moreover, if a semicategory lifts to a category, it does so in a unique way: the functor U:CatSemiCatU \colon Cat \to SemiCat is an injection on isomorphism classes.

Remark

Equivalently one could use the target map instead of the source map in the formulation of prop. 1.

Remark

The diagram appearing in prop. 1 is a simple version of the univalence condition appearing in definition of a complete semi-Segal space, a semi-category object in an (infinity,1)-category. See there for more on this.

Proposition

The functor UU of def. 3 has a left adjoint, which freely adjoins identity morphisms to a semicategory in the obvious way. It also has a right adjoint, which sends a semicategory SS to the category whose objects are the idempotents of SS and whose morphisms are the morphisms of SS that commute suitably with them, as described at Karoubi envelope. Indeed, the monad on Cat generated by this latter adjunction is exactly the monad for idempotent completion, also called Cauchy completion. (Note, however, that this is not a 2-monad, because the right adjoint of UU is not a 2-functor.)

Nerves and semi-simplicial sets

The nerve of a semicategory is a semi-simplicial set which satisfies the Segal conditions.

Examples

Start with the category of metric spaces and short maps. An occasionally useful semicategory can be formed from it by considering the nonempty spaces and strictly contractive functions.

This is a semicategory, since:

  • the composition of two strictly contractive functions is strictly contractive
  • identity maps are not contractive (they are trivial isometries)

The interest in this semicategory arises from the fact that all morphisms f:AAf : A \to A have unique fixed points, by Banach’s fixed point theorem.

In higher category theory

The concept of semicategory has more or less evident analogs and generalizations in higher category theory.

For models of higher categories by simplicial sets, i.e. presehaves on the simplex category (such as Kan complexes, quasi-categories, weak complicial sets) the corresponding semi-category notion is obtained by discarding the degeneracy maps (which are the identity-assigning maps in the simplicial framework), i.e. by considering just presheaves on the subcategory Δ +Δ\Delta_+ \subset \Delta on injective morphisms (see the discuss of Δ +\Delta_+ at Reedy model structure for more details).

Accordingly, there is the semi-category analog of a Segal space, called a semi-Segal space.

Simpson's conjecture says that every \infty-category has a model where all composition operations are strict and only the unit laws hold up to coherent homotopies. This would mean that the \infty-semicategory underlying any \infty-category can always be chosen to be strict.

References

Semicategories and semigroups are mentioned for instance in section 2 in

  • W. Dale Garraway, Sheaves for an involutive quantaloid, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 4 (2005), p. 243-274 (numdam)

Semicategories with an eye towards their generalization to semi-Segal spaces are briefly discussed at the beginning of

Revised on April 28, 2014 05:04:20 by Anonymous Coward (82.37.192.154)